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Theorem rngohomf 26476
Description: A ring homomorphism is a function. (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypotheses
Ref Expression
rnghomf.1  |-  G  =  ( 1st `  R
)
rnghomf.2  |-  X  =  ran  G
rnghomf.3  |-  J  =  ( 1st `  S
)
rnghomf.4  |-  Y  =  ran  J
Assertion
Ref Expression
rngohomf  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  F : X
--> Y )

Proof of Theorem rngohomf
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rnghomf.1 . . . . 5  |-  G  =  ( 1st `  R
)
2 eqid 2408 . . . . 5  |-  ( 2nd `  R )  =  ( 2nd `  R )
3 rnghomf.2 . . . . 5  |-  X  =  ran  G
4 eqid 2408 . . . . 5  |-  (GId `  ( 2nd `  R ) )  =  (GId `  ( 2nd `  R ) )
5 rnghomf.3 . . . . 5  |-  J  =  ( 1st `  S
)
6 eqid 2408 . . . . 5  |-  ( 2nd `  S )  =  ( 2nd `  S )
7 rnghomf.4 . . . . 5  |-  Y  =  ran  J
8 eqid 2408 . . . . 5  |-  (GId `  ( 2nd `  S ) )  =  (GId `  ( 2nd `  S ) )
91, 2, 3, 4, 5, 6, 7, 8isrngohom 26475 . . . 4  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( F  e.  ( R  RngHom  S )  <->  ( F : X --> Y  /\  ( F `  (GId `  ( 2nd `  R ) ) )  =  (GId `  ( 2nd `  S ) )  /\  A. x  e.  X  A. y  e.  X  ( ( F `  ( x G y ) )  =  ( ( F `
 x ) J ( F `  y
) )  /\  ( F `  ( x
( 2nd `  R
) y ) )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  y
) ) ) ) ) )
109biimpa 471 . . 3  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  F  e.  ( R  RngHom  S ) )  -> 
( F : X --> Y  /\  ( F `  (GId `  ( 2nd `  R
) ) )  =  (GId `  ( 2nd `  S ) )  /\  A. x  e.  X  A. y  e.  X  (
( F `  (
x G y ) )  =  ( ( F `  x ) J ( F `  y ) )  /\  ( F `  ( x ( 2nd `  R
) y ) )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  y
) ) ) ) )
1110simp1d 969 . 2  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  F  e.  ( R  RngHom  S ) )  ->  F : X --> Y )
12113impa 1148 1  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  F : X
--> Y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2670   ran crn 4842   -->wf 5413   ` cfv 5417  (class class class)co 6044   1stc1st 6310   2ndc2nd 6311  GIdcgi 21732   RingOpscrngo 21920    RngHom crnghom 26470
This theorem is referenced by:  rngohomcl  26477  rngogrphom  26481  rngohomco  26484  keridl  26536
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-map 6983  df-rngohom 26473
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