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Theorem rngoisocnv 26597
Description: The inverse of a ring isomorphism is a ring isomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)
Assertion
Ref Expression
rngoisocnv  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngIso  S ) )  ->  `' F  e.  ( S  RngIso  R ) )

Proof of Theorem rngoisocnv
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1ocnv 5687 . . . . . . . 8  |-  ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
)  ->  `' F : ran  ( 1st `  S
)
-1-1-onto-> ran  ( 1st `  R
) )
2 f1of 5674 . . . . . . . 8  |-  ( `' F : ran  ( 1st `  S ) -1-1-onto-> ran  ( 1st `  R )  ->  `' F : ran  ( 1st `  S ) --> ran  ( 1st `  R
) )
31, 2syl 16 . . . . . . 7  |-  ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
)  ->  `' F : ran  ( 1st `  S
) --> ran  ( 1st `  R ) )
43ad2antll 710 . . . . . 6  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  ->  `' F : ran  ( 1st `  S ) --> ran  ( 1st `  R
) )
5 eqid 2436 . . . . . . . . . 10  |-  ( 2nd `  R )  =  ( 2nd `  R )
6 eqid 2436 . . . . . . . . . 10  |-  (GId `  ( 2nd `  R ) )  =  (GId `  ( 2nd `  R ) )
7 eqid 2436 . . . . . . . . . 10  |-  ( 2nd `  S )  =  ( 2nd `  S )
8 eqid 2436 . . . . . . . . . 10  |-  (GId `  ( 2nd `  S ) )  =  (GId `  ( 2nd `  S ) )
95, 6, 7, 8rngohom1 26584 . . . . . . . . 9  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( F `  (GId `  ( 2nd `  R ) ) )  =  (GId `  ( 2nd `  S ) ) )
1093expa 1153 . . . . . . . 8  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  F  e.  ( R  RngHom  S ) )  -> 
( F `  (GId `  ( 2nd `  R
) ) )  =  (GId `  ( 2nd `  S ) ) )
1110adantrr 698 . . . . . . 7  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  -> 
( F `  (GId `  ( 2nd `  R
) ) )  =  (GId `  ( 2nd `  S ) ) )
12 eqid 2436 . . . . . . . . . . 11  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
1312, 5, 6rngo1cl 22017 . . . . . . . . . 10  |-  ( R  e.  RingOps  ->  (GId `  ( 2nd `  R ) )  e.  ran  ( 1st `  R ) )
14 f1ocnvfv 6016 . . . . . . . . . 10  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  (GId `  ( 2nd `  R
) )  e.  ran  ( 1st `  R ) )  ->  ( ( F `  (GId `  ( 2nd `  R ) ) )  =  (GId `  ( 2nd `  S ) )  ->  ( `' F `  (GId `  ( 2nd `  S ) ) )  =  (GId `  ( 2nd `  R ) ) ) )
1513, 14sylan2 461 . . . . . . . . 9  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  R  e.  RingOps )  -> 
( ( F `  (GId `  ( 2nd `  R
) ) )  =  (GId `  ( 2nd `  S ) )  -> 
( `' F `  (GId `  ( 2nd `  S
) ) )  =  (GId `  ( 2nd `  R ) ) ) )
1615ancoms 440 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
) )  ->  (
( F `  (GId `  ( 2nd `  R
) ) )  =  (GId `  ( 2nd `  S ) )  -> 
( `' F `  (GId `  ( 2nd `  S
) ) )  =  (GId `  ( 2nd `  R ) ) ) )
1716ad2ant2rl 730 . . . . . . 7  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  -> 
( ( F `  (GId `  ( 2nd `  R
) ) )  =  (GId `  ( 2nd `  S ) )  -> 
( `' F `  (GId `  ( 2nd `  S
) ) )  =  (GId `  ( 2nd `  R ) ) ) )
1811, 17mpd 15 . . . . . 6  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  -> 
( `' F `  (GId `  ( 2nd `  S
) ) )  =  (GId `  ( 2nd `  R ) ) )
19 f1ocnvfv2 6015 . . . . . . . . . . . . . 14  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  x  e.  ran  ( 1st `  S ) )  -> 
( F `  ( `' F `  x ) )  =  x )
20 f1ocnvfv2 6015 . . . . . . . . . . . . . 14  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) )  -> 
( F `  ( `' F `  y ) )  =  y )
2119, 20anim12da 26412 . . . . . . . . . . . . 13  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  (
( F `  ( `' F `  x ) )  =  x  /\  ( F `  ( `' F `  y ) )  =  y ) )
22 oveq12 6090 . . . . . . . . . . . . 13  |-  ( ( ( F `  ( `' F `  x ) )  =  x  /\  ( F `  ( `' F `  y ) )  =  y )  ->  ( ( F `
 ( `' F `  x ) ) ( 1st `  S ) ( F `  ( `' F `  y ) ) )  =  ( x ( 1st `  S
) y ) )
2321, 22syl 16 . . . . . . . . . . . 12  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  (
( F `  ( `' F `  x ) ) ( 1st `  S
) ( F `  ( `' F `  y ) ) )  =  ( x ( 1st `  S
) y ) )
2423adantll 695 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( ( F `
 ( `' F `  x ) ) ( 1st `  S ) ( F `  ( `' F `  y ) ) )  =  ( x ( 1st `  S
) y ) )
2524adantll 695 . . . . . . . . . 10  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  (
( F `  ( `' F `  x ) ) ( 1st `  S
) ( F `  ( `' F `  y ) ) )  =  ( x ( 1st `  S
) y ) )
26 f1ocnvdm 6018 . . . . . . . . . . . . . . . 16  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  x  e.  ran  ( 1st `  S ) )  -> 
( `' F `  x )  e.  ran  ( 1st `  R ) )
27 f1ocnvdm 6018 . . . . . . . . . . . . . . . 16  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) )  -> 
( `' F `  y )  e.  ran  ( 1st `  R ) )
2826, 27anim12da 26412 . . . . . . . . . . . . . . 15  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  (
( `' F `  x )  e.  ran  ( 1st `  R )  /\  ( `' F `  y )  e.  ran  ( 1st `  R ) ) )
29 eqid 2436 . . . . . . . . . . . . . . . 16  |-  ( 1st `  R )  =  ( 1st `  R )
30 eqid 2436 . . . . . . . . . . . . . . . 16  |-  ( 1st `  S )  =  ( 1st `  S )
3129, 12, 30rngohomadd 26585 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( ( `' F `  x )  e.  ran  ( 1st `  R )  /\  ( `' F `  y )  e.  ran  ( 1st `  R ) ) )  ->  ( F `  ( ( `' F `  x ) ( 1st `  R ) ( `' F `  y ) ) )  =  ( ( F `  ( `' F `  x ) ) ( 1st `  S
) ( F `  ( `' F `  y ) ) ) )
3228, 31sylan2 461 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
)  /\  ( x  e.  ran  ( 1st `  S
)  /\  y  e.  ran  ( 1st `  S
) ) ) )  ->  ( F `  ( ( `' F `  x ) ( 1st `  R ) ( `' F `  y ) ) )  =  ( ( F `  ( `' F `  x ) ) ( 1st `  S
) ( F `  ( `' F `  y ) ) ) )
3332exp32 589 . . . . . . . . . . . . 13  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
)  ->  ( (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) )  -> 
( F `  (
( `' F `  x ) ( 1st `  R ) ( `' F `  y ) ) )  =  ( ( F `  ( `' F `  x ) ) ( 1st `  S
) ( F `  ( `' F `  y ) ) ) ) ) )
34333expa 1153 . . . . . . . . . . . 12  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  F  e.  ( R  RngHom  S ) )  -> 
( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
)  ->  ( (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) )  -> 
( F `  (
( `' F `  x ) ( 1st `  R ) ( `' F `  y ) ) )  =  ( ( F `  ( `' F `  x ) ) ( 1st `  S
) ( F `  ( `' F `  y ) ) ) ) ) )
3534impr 603 . . . . . . . . . . 11  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  -> 
( ( x  e. 
ran  ( 1st `  S
)  /\  y  e.  ran  ( 1st `  S
) )  ->  ( F `  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) ) )  =  ( ( F `
 ( `' F `  x ) ) ( 1st `  S ) ( F `  ( `' F `  y ) ) ) ) )
3635imp 419 . . . . . . . . . 10  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( F `  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) ) )  =  ( ( F `
 ( `' F `  x ) ) ( 1st `  S ) ( F `  ( `' F `  y ) ) ) )
37 eqid 2436 . . . . . . . . . . . . . . . 16  |-  ran  ( 1st `  S )  =  ran  ( 1st `  S
)
3830, 37rngogcl 21979 . . . . . . . . . . . . . . 15  |-  ( ( S  e.  RingOps  /\  x  e.  ran  ( 1st `  S
)  /\  y  e.  ran  ( 1st `  S
) )  ->  (
x ( 1st `  S
) y )  e. 
ran  ( 1st `  S
) )
39383expb 1154 . . . . . . . . . . . . . 14  |-  ( ( S  e.  RingOps  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( x ( 1st `  S ) y )  e.  ran  ( 1st `  S ) )
40 f1ocnvfv2 6015 . . . . . . . . . . . . . . 15  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  ( x ( 1st `  S ) y )  e.  ran  ( 1st `  S ) )  -> 
( F `  ( `' F `  ( x ( 1st `  S
) y ) ) )  =  ( x ( 1st `  S
) y ) )
4140ancoms 440 . . . . . . . . . . . . . 14  |-  ( ( ( x ( 1st `  S ) y )  e.  ran  ( 1st `  S )  /\  F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
) )  ->  ( F `  ( `' F `  ( x
( 1st `  S
) y ) ) )  =  ( x ( 1st `  S
) y ) )
4239, 41sylan 458 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  RingOps  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  /\  F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
) )  ->  ( F `  ( `' F `  ( x
( 1st `  S
) y ) ) )  =  ( x ( 1st `  S
) y ) )
4342an32s 780 . . . . . . . . . . . 12  |-  ( ( ( S  e.  RingOps  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( F `  ( `' F `  ( x ( 1st `  S
) y ) ) )  =  ( x ( 1st `  S
) y ) )
4443adantlll 699 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( F `  ( `' F `  ( x ( 1st `  S
) y ) ) )  =  ( x ( 1st `  S
) y ) )
4544adantlrl 701 . . . . . . . . . 10  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( F `  ( `' F `  ( x
( 1st `  S
) y ) ) )  =  ( x ( 1st `  S
) y ) )
4625, 36, 453eqtr4rd 2479 . . . . . . . . 9  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( F `  ( `' F `  ( x
( 1st `  S
) y ) ) )  =  ( F `
 ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) ) ) )
47 f1of1 5673 . . . . . . . . . . . 12  |-  ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
)  ->  F : ran  ( 1st `  R
) -1-1-> ran  ( 1st `  S
) )
4847ad2antlr 708 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  F : ran  ( 1st `  R )
-1-1-> ran  ( 1st `  S
) )
49 f1ocnvdm 6018 . . . . . . . . . . . . . . 15  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  ( x ( 1st `  S ) y )  e.  ran  ( 1st `  S ) )  -> 
( `' F `  ( x ( 1st `  S ) y ) )  e.  ran  ( 1st `  R ) )
5049ancoms 440 . . . . . . . . . . . . . 14  |-  ( ( ( x ( 1st `  S ) y )  e.  ran  ( 1st `  S )  /\  F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
) )  ->  ( `' F `  ( x ( 1st `  S
) y ) )  e.  ran  ( 1st `  R ) )
5139, 50sylan 458 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  RingOps  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  /\  F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
) )  ->  ( `' F `  ( x ( 1st `  S
) y ) )  e.  ran  ( 1st `  R ) )
5251an32s 780 . . . . . . . . . . . 12  |-  ( ( ( S  e.  RingOps  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( `' F `  ( x ( 1st `  S ) y ) )  e.  ran  ( 1st `  R ) )
5352adantlll 699 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( `' F `  ( x ( 1st `  S ) y ) )  e.  ran  ( 1st `  R ) )
5429, 12rngogcl 21979 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  RingOps  /\  ( `' F `  x )  e.  ran  ( 1st `  R )  /\  ( `' F `  y )  e.  ran  ( 1st `  R ) )  -> 
( ( `' F `  x ) ( 1st `  R ) ( `' F `  y ) )  e.  ran  ( 1st `  R ) )
55543expb 1154 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RingOps  /\  (
( `' F `  x )  e.  ran  ( 1st `  R )  /\  ( `' F `  y )  e.  ran  ( 1st `  R ) ) )  ->  (
( `' F `  x ) ( 1st `  R ) ( `' F `  y ) )  e.  ran  ( 1st `  R ) )
5628, 55sylan2 461 . . . . . . . . . . . . 13  |-  ( ( R  e.  RingOps  /\  ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
)  /\  ( x  e.  ran  ( 1st `  S
)  /\  y  e.  ran  ( 1st `  S
) ) ) )  ->  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) )  e. 
ran  ( 1st `  R
) )
5756anassrs 630 . . . . . . . . . . . 12  |-  ( ( ( R  e.  RingOps  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) )  e. 
ran  ( 1st `  R
) )
5857adantllr 700 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) )  e. 
ran  ( 1st `  R
) )
59 f1fveq 6008 . . . . . . . . . . 11  |-  ( ( F : ran  ( 1st `  R ) -1-1-> ran  ( 1st `  S )  /\  ( ( `' F `  ( x ( 1st `  S
) y ) )  e.  ran  ( 1st `  R )  /\  (
( `' F `  x ) ( 1st `  R ) ( `' F `  y ) )  e.  ran  ( 1st `  R ) ) )  ->  ( ( F `  ( `' F `  ( x
( 1st `  S
) y ) ) )  =  ( F `
 ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) ) )  <-> 
( `' F `  ( x ( 1st `  S ) y ) )  =  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) ) ) )
6048, 53, 58, 59syl12anc 1182 . . . . . . . . . 10  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( ( F `
 ( `' F `  ( x ( 1st `  S ) y ) ) )  =  ( F `  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) ) )  <-> 
( `' F `  ( x ( 1st `  S ) y ) )  =  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) ) ) )
6160adantlrl 701 . . . . . . . . 9  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  (
( F `  ( `' F `  ( x ( 1st `  S
) y ) ) )  =  ( F `
 ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) ) )  <-> 
( `' F `  ( x ( 1st `  S ) y ) )  =  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) ) ) )
6246, 61mpbid 202 . . . . . . . 8  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( `' F `  ( x ( 1st `  S
) y ) )  =  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) ) )
63 oveq12 6090 . . . . . . . . . . . . 13  |-  ( ( ( F `  ( `' F `  x ) )  =  x  /\  ( F `  ( `' F `  y ) )  =  y )  ->  ( ( F `
 ( `' F `  x ) ) ( 2nd `  S ) ( F `  ( `' F `  y ) ) )  =  ( x ( 2nd `  S
) y ) )
6421, 63syl 16 . . . . . . . . . . . 12  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  (
( F `  ( `' F `  x ) ) ( 2nd `  S
) ( F `  ( `' F `  y ) ) )  =  ( x ( 2nd `  S
) y ) )
6564adantll 695 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( ( F `
 ( `' F `  x ) ) ( 2nd `  S ) ( F `  ( `' F `  y ) ) )  =  ( x ( 2nd `  S
) y ) )
6665adantll 695 . . . . . . . . . 10  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  (
( F `  ( `' F `  x ) ) ( 2nd `  S
) ( F `  ( `' F `  y ) ) )  =  ( x ( 2nd `  S
) y ) )
6729, 12, 5, 7rngohommul 26586 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( ( `' F `  x )  e.  ran  ( 1st `  R )  /\  ( `' F `  y )  e.  ran  ( 1st `  R ) ) )  ->  ( F `  ( ( `' F `  x ) ( 2nd `  R ) ( `' F `  y ) ) )  =  ( ( F `  ( `' F `  x ) ) ( 2nd `  S
) ( F `  ( `' F `  y ) ) ) )
6828, 67sylan2 461 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
)  /\  ( x  e.  ran  ( 1st `  S
)  /\  y  e.  ran  ( 1st `  S
) ) ) )  ->  ( F `  ( ( `' F `  x ) ( 2nd `  R ) ( `' F `  y ) ) )  =  ( ( F `  ( `' F `  x ) ) ( 2nd `  S
) ( F `  ( `' F `  y ) ) ) )
6968exp32 589 . . . . . . . . . . . . 13  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
)  ->  ( (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) )  -> 
( F `  (
( `' F `  x ) ( 2nd `  R ) ( `' F `  y ) ) )  =  ( ( F `  ( `' F `  x ) ) ( 2nd `  S
) ( F `  ( `' F `  y ) ) ) ) ) )
70693expa 1153 . . . . . . . . . . . 12  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  F  e.  ( R  RngHom  S ) )  -> 
( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
)  ->  ( (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) )  -> 
( F `  (
( `' F `  x ) ( 2nd `  R ) ( `' F `  y ) ) )  =  ( ( F `  ( `' F `  x ) ) ( 2nd `  S
) ( F `  ( `' F `  y ) ) ) ) ) )
7170impr 603 . . . . . . . . . . 11  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  -> 
( ( x  e. 
ran  ( 1st `  S
)  /\  y  e.  ran  ( 1st `  S
) )  ->  ( F `  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) )  =  ( ( F `
 ( `' F `  x ) ) ( 2nd `  S ) ( F `  ( `' F `  y ) ) ) ) )
7271imp 419 . . . . . . . . . 10  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( F `  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) )  =  ( ( F `
 ( `' F `  x ) ) ( 2nd `  S ) ( F `  ( `' F `  y ) ) ) )
7330, 7, 37rngocl 21970 . . . . . . . . . . . . . . 15  |-  ( ( S  e.  RingOps  /\  x  e.  ran  ( 1st `  S
)  /\  y  e.  ran  ( 1st `  S
) )  ->  (
x ( 2nd `  S
) y )  e. 
ran  ( 1st `  S
) )
74733expb 1154 . . . . . . . . . . . . . 14  |-  ( ( S  e.  RingOps  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( x ( 2nd `  S ) y )  e.  ran  ( 1st `  S ) )
75 f1ocnvfv2 6015 . . . . . . . . . . . . . . 15  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  ( x ( 2nd `  S ) y )  e.  ran  ( 1st `  S ) )  -> 
( F `  ( `' F `  ( x ( 2nd `  S
) y ) ) )  =  ( x ( 2nd `  S
) y ) )
7675ancoms 440 . . . . . . . . . . . . . 14  |-  ( ( ( x ( 2nd `  S ) y )  e.  ran  ( 1st `  S )  /\  F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
) )  ->  ( F `  ( `' F `  ( x
( 2nd `  S
) y ) ) )  =  ( x ( 2nd `  S
) y ) )
7774, 76sylan 458 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  RingOps  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  /\  F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
) )  ->  ( F `  ( `' F `  ( x
( 2nd `  S
) y ) ) )  =  ( x ( 2nd `  S
) y ) )
7877an32s 780 . . . . . . . . . . . 12  |-  ( ( ( S  e.  RingOps  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( F `  ( `' F `  ( x ( 2nd `  S
) y ) ) )  =  ( x ( 2nd `  S
) y ) )
7978adantlll 699 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( F `  ( `' F `  ( x ( 2nd `  S
) y ) ) )  =  ( x ( 2nd `  S
) y ) )
8079adantlrl 701 . . . . . . . . . 10  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( F `  ( `' F `  ( x
( 2nd `  S
) y ) ) )  =  ( x ( 2nd `  S
) y ) )
8166, 72, 803eqtr4rd 2479 . . . . . . . . 9  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( F `  ( `' F `  ( x
( 2nd `  S
) y ) ) )  =  ( F `
 ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) ) )
82 f1ocnvdm 6018 . . . . . . . . . . . . . . 15  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  ( x ( 2nd `  S ) y )  e.  ran  ( 1st `  S ) )  -> 
( `' F `  ( x ( 2nd `  S ) y ) )  e.  ran  ( 1st `  R ) )
8382ancoms 440 . . . . . . . . . . . . . 14  |-  ( ( ( x ( 2nd `  S ) y )  e.  ran  ( 1st `  S )  /\  F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
) )  ->  ( `' F `  ( x ( 2nd `  S
) y ) )  e.  ran  ( 1st `  R ) )
8474, 83sylan 458 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  RingOps  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  /\  F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
) )  ->  ( `' F `  ( x ( 2nd `  S
) y ) )  e.  ran  ( 1st `  R ) )
8584an32s 780 . . . . . . . . . . . 12  |-  ( ( ( S  e.  RingOps  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( `' F `  ( x ( 2nd `  S ) y ) )  e.  ran  ( 1st `  R ) )
8685adantlll 699 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( `' F `  ( x ( 2nd `  S ) y ) )  e.  ran  ( 1st `  R ) )
8729, 5, 12rngocl 21970 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  RingOps  /\  ( `' F `  x )  e.  ran  ( 1st `  R )  /\  ( `' F `  y )  e.  ran  ( 1st `  R ) )  -> 
( ( `' F `  x ) ( 2nd `  R ) ( `' F `  y ) )  e.  ran  ( 1st `  R ) )
88873expb 1154 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RingOps  /\  (
( `' F `  x )  e.  ran  ( 1st `  R )  /\  ( `' F `  y )  e.  ran  ( 1st `  R ) ) )  ->  (
( `' F `  x ) ( 2nd `  R ) ( `' F `  y ) )  e.  ran  ( 1st `  R ) )
8928, 88sylan2 461 . . . . . . . . . . . . 13  |-  ( ( R  e.  RingOps  /\  ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
)  /\  ( x  e.  ran  ( 1st `  S
)  /\  y  e.  ran  ( 1st `  S
) ) ) )  ->  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) )  e. 
ran  ( 1st `  R
) )
9089anassrs 630 . . . . . . . . . . . 12  |-  ( ( ( R  e.  RingOps  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) )  e. 
ran  ( 1st `  R
) )
9190adantllr 700 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) )  e. 
ran  ( 1st `  R
) )
92 f1fveq 6008 . . . . . . . . . . 11  |-  ( ( F : ran  ( 1st `  R ) -1-1-> ran  ( 1st `  S )  /\  ( ( `' F `  ( x ( 2nd `  S
) y ) )  e.  ran  ( 1st `  R )  /\  (
( `' F `  x ) ( 2nd `  R ) ( `' F `  y ) )  e.  ran  ( 1st `  R ) ) )  ->  ( ( F `  ( `' F `  ( x
( 2nd `  S
) y ) ) )  =  ( F `
 ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) )  <-> 
( `' F `  ( x ( 2nd `  S ) y ) )  =  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) ) )
9348, 86, 91, 92syl12anc 1182 . . . . . . . . . 10  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( ( F `
 ( `' F `  ( x ( 2nd `  S ) y ) ) )  =  ( F `  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) )  <-> 
( `' F `  ( x ( 2nd `  S ) y ) )  =  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) ) )
9493adantlrl 701 . . . . . . . . 9  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  (
( F `  ( `' F `  ( x ( 2nd `  S
) y ) ) )  =  ( F `
 ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) )  <-> 
( `' F `  ( x ( 2nd `  S ) y ) )  =  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) ) )
9581, 94mpbid 202 . . . . . . . 8  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( `' F `  ( x ( 2nd `  S
) y ) )  =  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) )
9662, 95jca 519 . . . . . . 7  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  (
( `' F `  ( x ( 1st `  S ) y ) )  =  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) )  /\  ( `' F `  ( x ( 2nd `  S
) y ) )  =  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) ) )
9796ralrimivva 2798 . . . . . 6  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  ->  A. x  e.  ran  ( 1st `  S ) A. y  e.  ran  ( 1st `  S ) ( ( `' F `  ( x ( 1st `  S ) y ) )  =  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) )  /\  ( `' F `  ( x ( 2nd `  S
) y ) )  =  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) ) )
9830, 7, 37, 8, 29, 5, 12, 6isrngohom 26581 . . . . . . . 8  |-  ( ( S  e.  RingOps  /\  R  e.  RingOps )  ->  ( `' F  e.  ( S  RngHom  R )  <->  ( `' F : ran  ( 1st `  S ) --> ran  ( 1st `  R )  /\  ( `' F `  (GId `  ( 2nd `  S ) ) )  =  (GId
`  ( 2nd `  R
) )  /\  A. x  e.  ran  ( 1st `  S ) A. y  e.  ran  ( 1st `  S
) ( ( `' F `  ( x ( 1st `  S
) y ) )  =  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) )  /\  ( `' F `  ( x ( 2nd `  S
) y ) )  =  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) ) ) ) )
9998ancoms 440 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( `' F  e.  ( S  RngHom  R )  <->  ( `' F : ran  ( 1st `  S ) --> ran  ( 1st `  R )  /\  ( `' F `  (GId `  ( 2nd `  S ) ) )  =  (GId
`  ( 2nd `  R
) )  /\  A. x  e.  ran  ( 1st `  S ) A. y  e.  ran  ( 1st `  S
) ( ( `' F `  ( x ( 1st `  S
) y ) )  =  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) )  /\  ( `' F `  ( x ( 2nd `  S
) y ) )  =  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) ) ) ) )
10099adantr 452 . . . . . 6  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  -> 
( `' F  e.  ( S  RngHom  R )  <-> 
( `' F : ran  ( 1st `  S
) --> ran  ( 1st `  R )  /\  ( `' F `  (GId `  ( 2nd `  S ) ) )  =  (GId
`  ( 2nd `  R
) )  /\  A. x  e.  ran  ( 1st `  S ) A. y  e.  ran  ( 1st `  S
) ( ( `' F `  ( x ( 1st `  S
) y ) )  =  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) )  /\  ( `' F `  ( x ( 2nd `  S
) y ) )  =  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) ) ) ) )
1014, 18, 97, 100mpbir3and 1137 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  ->  `' F  e.  ( S  RngHom  R ) )
1021ad2antll 710 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  ->  `' F : ran  ( 1st `  S ) -1-1-onto-> ran  ( 1st `  R ) )
103101, 102jca 519 . . . 4  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  -> 
( `' F  e.  ( S  RngHom  R )  /\  `' F : ran  ( 1st `  S
)
-1-1-onto-> ran  ( 1st `  R
) ) )
104103ex 424 . . 3  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  (
( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  ->  ( `' F  e.  ( S  RngHom  R )  /\  `' F : ran  ( 1st `  S ) -1-1-onto-> ran  ( 1st `  R ) ) ) )
10529, 12, 30, 37isrngoiso 26594 . . 3  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( F  e.  ( R  RngIso  S )  <->  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) ) )
10630, 37, 29, 12isrngoiso 26594 . . . 4  |-  ( ( S  e.  RingOps  /\  R  e.  RingOps )  ->  ( `' F  e.  ( S  RngIso  R )  <->  ( `' F  e.  ( S  RngHom  R )  /\  `' F : ran  ( 1st `  S ) -1-1-onto-> ran  ( 1st `  R
) ) ) )
107106ancoms 440 . . 3  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( `' F  e.  ( S  RngIso  R )  <->  ( `' F  e.  ( S  RngHom  R )  /\  `' F : ran  ( 1st `  S ) -1-1-onto-> ran  ( 1st `  R
) ) ) )
108104, 105, 1073imtr4d 260 . 2  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( F  e.  ( R  RngIso  S )  ->  `' F  e.  ( S  RngIso  R ) ) )
1091083impia 1150 1  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngIso  S ) )  ->  `' F  e.  ( S  RngIso  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   `'ccnv 4877   ran crn 4879   -->wf 5450   -1-1->wf1 5451   -1-1-onto->wf1o 5453   ` cfv 5454  (class class class)co 6081   1stc1st 6347   2ndc2nd 6348  GIdcgi 21775   RingOpscrngo 21963    RngHom crnghom 26576    RngIso crngiso 26577
This theorem is referenced by:  riscer  26604
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-map 7020  df-grpo 21779  df-gid 21780  df-ablo 21870  df-ass 21901  df-exid 21903  df-mgm 21907  df-sgr 21919  df-mndo 21926  df-rngo 21964  df-rngohom 26579  df-rngoiso 26592
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