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Theorem isofrlem 6052
Description: Lemma for isofr 6054. (Contributed by NM, 29-Apr-2004.) (Revised by Mario Carneiro, 18-Nov-2014.)
Hypotheses
Ref Expression
isofrlem.1  |-  ( ph  ->  H  Isom  R ,  S  ( A ,  B ) )
isofrlem.2  |-  ( ph  ->  ( H " x
)  e.  _V )
Assertion
Ref Expression
isofrlem  |-  ( ph  ->  ( S  Fr  B  ->  R  Fr  A ) )
Distinct variable groups:    x, A    x, B    x, H    ph, x    x, R    x, S

Proof of Theorem isofrlem
Dummy variables  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isofrlem.1 . . . . . . 7  |-  ( ph  ->  H  Isom  R ,  S  ( A ,  B ) )
2 isof1o 6037 . . . . . . 7  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)
31, 2syl 16 . . . . . 6  |-  ( ph  ->  H : A -1-1-onto-> B )
4 f1ofn 5667 . . . . . . . 8  |-  ( H : A -1-1-onto-> B  ->  H  Fn  A )
5 n0 3629 . . . . . . . . . 10  |-  ( x  =/=  (/)  <->  E. y  y  e.  x )
6 fnfvima 5968 . . . . . . . . . . . . 13  |-  ( ( H  Fn  A  /\  x  C_  A  /\  y  e.  x )  ->  ( H `  y )  e.  ( H " x
) )
7 ne0i 3626 . . . . . . . . . . . . 13  |-  ( ( H `  y )  e.  ( H "
x )  ->  ( H " x )  =/=  (/) )
86, 7syl 16 . . . . . . . . . . . 12  |-  ( ( H  Fn  A  /\  x  C_  A  /\  y  e.  x )  ->  ( H " x )  =/=  (/) )
983expia 1155 . . . . . . . . . . 11  |-  ( ( H  Fn  A  /\  x  C_  A )  -> 
( y  e.  x  ->  ( H " x
)  =/=  (/) ) )
109exlimdv 1646 . . . . . . . . . 10  |-  ( ( H  Fn  A  /\  x  C_  A )  -> 
( E. y  y  e.  x  ->  ( H " x )  =/=  (/) ) )
115, 10syl5bi 209 . . . . . . . . 9  |-  ( ( H  Fn  A  /\  x  C_  A )  -> 
( x  =/=  (/)  ->  ( H " x )  =/=  (/) ) )
1211expimpd 587 . . . . . . . 8  |-  ( H  Fn  A  ->  (
( x  C_  A  /\  x  =/=  (/) )  -> 
( H " x
)  =/=  (/) ) )
134, 12syl 16 . . . . . . 7  |-  ( H : A -1-1-onto-> B  ->  ( (
x  C_  A  /\  x  =/=  (/) )  ->  ( H " x )  =/=  (/) ) )
14 f1ofo 5673 . . . . . . . 8  |-  ( H : A -1-1-onto-> B  ->  H : A -onto-> B )
15 imassrn 5208 . . . . . . . . 9  |-  ( H
" x )  C_  ran  H
16 forn 5648 . . . . . . . . 9  |-  ( H : A -onto-> B  ->  ran  H  =  B )
1715, 16syl5sseq 3388 . . . . . . . 8  |-  ( H : A -onto-> B  -> 
( H " x
)  C_  B )
1814, 17syl 16 . . . . . . 7  |-  ( H : A -1-1-onto-> B  ->  ( H " x )  C_  B
)
1913, 18jctild 528 . . . . . 6  |-  ( H : A -1-1-onto-> B  ->  ( (
x  C_  A  /\  x  =/=  (/) )  ->  (
( H " x
)  C_  B  /\  ( H " x )  =/=  (/) ) ) )
203, 19syl 16 . . . . 5  |-  ( ph  ->  ( ( x  C_  A  /\  x  =/=  (/) )  -> 
( ( H "
x )  C_  B  /\  ( H " x
)  =/=  (/) ) ) )
21 dffr3 5228 . . . . . 6  |-  ( S  Fr  B  <->  A. z
( ( z  C_  B  /\  z  =/=  (/) )  ->  E. w  e.  z 
( z  i^i  ( `' S " { w } ) )  =  (/) ) )
22 isofrlem.2 . . . . . . 7  |-  ( ph  ->  ( H " x
)  e.  _V )
23 sseq1 3361 . . . . . . . . . 10  |-  ( z  =  ( H "
x )  ->  (
z  C_  B  <->  ( H " x )  C_  B
) )
24 neeq1 2606 . . . . . . . . . 10  |-  ( z  =  ( H "
x )  ->  (
z  =/=  (/)  <->  ( H " x )  =/=  (/) ) )
2523, 24anbi12d 692 . . . . . . . . 9  |-  ( z  =  ( H "
x )  ->  (
( z  C_  B  /\  z  =/=  (/) )  <->  ( ( H " x )  C_  B  /\  ( H "
x )  =/=  (/) ) ) )
26 ineq1 3527 . . . . . . . . . . 11  |-  ( z  =  ( H "
x )  ->  (
z  i^i  ( `' S " { w }
) )  =  ( ( H " x
)  i^i  ( `' S " { w }
) ) )
2726eqeq1d 2443 . . . . . . . . . 10  |-  ( z  =  ( H "
x )  ->  (
( z  i^i  ( `' S " { w } ) )  =  (/) 
<->  ( ( H "
x )  i^i  ( `' S " { w } ) )  =  (/) ) )
2827rexeqbi1dv 2905 . . . . . . . . 9  |-  ( z  =  ( H "
x )  ->  ( E. w  e.  z 
( z  i^i  ( `' S " { w } ) )  =  (/) 
<->  E. w  e.  ( H " x ) ( ( H "
x )  i^i  ( `' S " { w } ) )  =  (/) ) )
2925, 28imbi12d 312 . . . . . . . 8  |-  ( z  =  ( H "
x )  ->  (
( ( z  C_  B  /\  z  =/=  (/) )  ->  E. w  e.  z 
( z  i^i  ( `' S " { w } ) )  =  (/) )  <->  ( ( ( H " x ) 
C_  B  /\  ( H " x )  =/=  (/) )  ->  E. w  e.  ( H " x
) ( ( H
" x )  i^i  ( `' S " { w } ) )  =  (/) ) ) )
3029spcgv 3028 . . . . . . 7  |-  ( ( H " x )  e.  _V  ->  ( A. z ( ( z 
C_  B  /\  z  =/=  (/) )  ->  E. w  e.  z  ( z  i^i  ( `' S " { w } ) )  =  (/) )  -> 
( ( ( H
" x )  C_  B  /\  ( H "
x )  =/=  (/) )  ->  E. w  e.  ( H " x ) ( ( H " x
)  i^i  ( `' S " { w }
) )  =  (/) ) ) )
3122, 30syl 16 . . . . . 6  |-  ( ph  ->  ( A. z ( ( z  C_  B  /\  z  =/=  (/) )  ->  E. w  e.  z 
( z  i^i  ( `' S " { w } ) )  =  (/) )  ->  ( ( ( H " x
)  C_  B  /\  ( H " x )  =/=  (/) )  ->  E. w  e.  ( H " x
) ( ( H
" x )  i^i  ( `' S " { w } ) )  =  (/) ) ) )
3221, 31syl5bi 209 . . . . 5  |-  ( ph  ->  ( S  Fr  B  ->  ( ( ( H
" x )  C_  B  /\  ( H "
x )  =/=  (/) )  ->  E. w  e.  ( H " x ) ( ( H " x
)  i^i  ( `' S " { w }
) )  =  (/) ) ) )
3320, 32syl5d 64 . . . 4  |-  ( ph  ->  ( S  Fr  B  ->  ( ( x  C_  A  /\  x  =/=  (/) )  ->  E. w  e.  ( H " x ) ( ( H " x
)  i^i  ( `' S " { w }
) )  =  (/) ) ) )
343adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  x  C_  A
)  ->  H : A
-1-1-onto-> B )
35 f1ofun 5668 . . . . . . . . . . 11  |-  ( H : A -1-1-onto-> B  ->  Fun  H )
3634, 35syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  x  C_  A
)  ->  Fun  H )
37 simpl 444 . . . . . . . . . 10  |-  ( ( w  e.  ( H
" x )  /\  ( ( H "
x )  i^i  ( `' S " { w } ) )  =  (/) )  ->  w  e.  ( H " x
) )
38 fvelima 5770 . . . . . . . . . 10  |-  ( ( Fun  H  /\  w  e.  ( H " x
) )  ->  E. y  e.  x  ( H `  y )  =  w )
3936, 37, 38syl2an 464 . . . . . . . . 9  |-  ( ( ( ph  /\  x  C_  A )  /\  (
w  e.  ( H
" x )  /\  ( ( H "
x )  i^i  ( `' S " { w } ) )  =  (/) ) )  ->  E. y  e.  x  ( H `  y )  =  w )
40 simpr 448 . . . . . . . . . . . . . . . 16  |-  ( ( w  e.  ( H
" x )  /\  ( ( H "
x )  i^i  ( `' S " { w } ) )  =  (/) )  ->  ( ( H " x )  i^i  ( `' S " { w } ) )  =  (/) )
41 ssel 3334 . . . . . . . . . . . . . . . . . . 19  |-  ( x 
C_  A  ->  (
y  e.  x  -> 
y  e.  A ) )
4241imdistani 672 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  C_  A  /\  y  e.  x )  ->  ( x  C_  A  /\  y  e.  A
) )
43 isomin 6049 . . . . . . . . . . . . . . . . . 18  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
x  C_  A  /\  y  e.  A )
)  ->  ( (
x  i^i  ( `' R " { y } ) )  =  (/)  <->  (
( H " x
)  i^i  ( `' S " { ( H `
 y ) } ) )  =  (/) ) )
441, 42, 43syl2an 464 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( x  C_  A  /\  y  e.  x ) )  -> 
( ( x  i^i  ( `' R " { y } ) )  =  (/)  <->  ( ( H " x )  i^i  ( `' S " { ( H `  y ) } ) )  =  (/) ) )
45 sneq 3817 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( H `  y )  =  w  ->  { ( H `  y ) }  =  { w } )
4645imaeq2d 5195 . . . . . . . . . . . . . . . . . . 19  |-  ( ( H `  y )  =  w  ->  ( `' S " { ( H `  y ) } )  =  ( `' S " { w } ) )
4746ineq2d 3534 . . . . . . . . . . . . . . . . . 18  |-  ( ( H `  y )  =  w  ->  (
( H " x
)  i^i  ( `' S " { ( H `
 y ) } ) )  =  ( ( H " x
)  i^i  ( `' S " { w }
) ) )
4847eqeq1d 2443 . . . . . . . . . . . . . . . . 17  |-  ( ( H `  y )  =  w  ->  (
( ( H "
x )  i^i  ( `' S " { ( H `  y ) } ) )  =  (/) 
<->  ( ( H "
x )  i^i  ( `' S " { w } ) )  =  (/) ) )
4944, 48sylan9bb 681 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  C_  A  /\  y  e.  x )
)  /\  ( H `  y )  =  w )  ->  ( (
x  i^i  ( `' R " { y } ) )  =  (/)  <->  (
( H " x
)  i^i  ( `' S " { w }
) )  =  (/) ) )
5040, 49syl5ibr 213 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  C_  A  /\  y  e.  x )
)  /\  ( H `  y )  =  w )  ->  ( (
w  e.  ( H
" x )  /\  ( ( H "
x )  i^i  ( `' S " { w } ) )  =  (/) )  ->  ( x  i^i  ( `' R " { y } ) )  =  (/) ) )
5150exp42 595 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( x  C_  A  ->  ( y  e.  x  ->  ( ( H `  y )  =  w  ->  ( ( w  e.  ( H "
x )  /\  (
( H " x
)  i^i  ( `' S " { w }
) )  =  (/) )  ->  ( x  i^i  ( `' R " { y } ) )  =  (/) ) ) ) ) )
5251imp 419 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  C_  A
)  ->  ( y  e.  x  ->  ( ( H `  y )  =  w  ->  (
( w  e.  ( H " x )  /\  ( ( H
" x )  i^i  ( `' S " { w } ) )  =  (/) )  -> 
( x  i^i  ( `' R " { y } ) )  =  (/) ) ) ) )
5352com3l 77 . . . . . . . . . . . 12  |-  ( y  e.  x  ->  (
( H `  y
)  =  w  -> 
( ( ph  /\  x  C_  A )  -> 
( ( w  e.  ( H " x
)  /\  ( ( H " x )  i^i  ( `' S " { w } ) )  =  (/) )  -> 
( x  i^i  ( `' R " { y } ) )  =  (/) ) ) ) )
5453com4t 81 . . . . . . . . . . 11  |-  ( (
ph  /\  x  C_  A
)  ->  ( (
w  e.  ( H
" x )  /\  ( ( H "
x )  i^i  ( `' S " { w } ) )  =  (/) )  ->  ( y  e.  x  ->  (
( H `  y
)  =  w  -> 
( x  i^i  ( `' R " { y } ) )  =  (/) ) ) ) )
5554imp 419 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  C_  A )  /\  (
w  e.  ( H
" x )  /\  ( ( H "
x )  i^i  ( `' S " { w } ) )  =  (/) ) )  ->  (
y  e.  x  -> 
( ( H `  y )  =  w  ->  ( x  i^i  ( `' R " { y } ) )  =  (/) ) ) )
5655reximdvai 2808 . . . . . . . . 9  |-  ( ( ( ph  /\  x  C_  A )  /\  (
w  e.  ( H
" x )  /\  ( ( H "
x )  i^i  ( `' S " { w } ) )  =  (/) ) )  ->  ( E. y  e.  x  ( H `  y )  =  w  ->  E. y  e.  x  ( x  i^i  ( `' R " { y } ) )  =  (/) ) )
5739, 56mpd 15 . . . . . . . 8  |-  ( ( ( ph  /\  x  C_  A )  /\  (
w  e.  ( H
" x )  /\  ( ( H "
x )  i^i  ( `' S " { w } ) )  =  (/) ) )  ->  E. y  e.  x  ( x  i^i  ( `' R " { y } ) )  =  (/) )
5857rexlimdvaa 2823 . . . . . . 7  |-  ( (
ph  /\  x  C_  A
)  ->  ( E. w  e.  ( H " x ) ( ( H " x )  i^i  ( `' S " { w } ) )  =  (/)  ->  E. y  e.  x  ( x  i^i  ( `' R " { y } ) )  =  (/) ) )
5958ex 424 . . . . . 6  |-  ( ph  ->  ( x  C_  A  ->  ( E. w  e.  ( H " x
) ( ( H
" x )  i^i  ( `' S " { w } ) )  =  (/)  ->  E. y  e.  x  ( x  i^i  ( `' R " { y } ) )  =  (/) ) ) )
6059adantrd 455 . . . . 5  |-  ( ph  ->  ( ( x  C_  A  /\  x  =/=  (/) )  -> 
( E. w  e.  ( H " x
) ( ( H
" x )  i^i  ( `' S " { w } ) )  =  (/)  ->  E. y  e.  x  ( x  i^i  ( `' R " { y } ) )  =  (/) ) ) )
6160a2d 24 . . . 4  |-  ( ph  ->  ( ( ( x 
C_  A  /\  x  =/=  (/) )  ->  E. w  e.  ( H " x
) ( ( H
" x )  i^i  ( `' S " { w } ) )  =  (/) )  -> 
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  ( x  i^i  ( `' R " { y } ) )  =  (/) ) ) )
6233, 61syld 42 . . 3  |-  ( ph  ->  ( S  Fr  B  ->  ( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  ( x  i^i  ( `' R " { y } ) )  =  (/) ) ) )
6362alrimdv 1643 . 2  |-  ( ph  ->  ( S  Fr  B  ->  A. x ( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  ( x  i^i  ( `' R " { y } ) )  =  (/) ) ) )
64 dffr3 5228 . 2  |-  ( R  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  ( x  i^i  ( `' R " { y } ) )  =  (/) ) )
6563, 64syl6ibr 219 1  |-  ( ph  ->  ( S  Fr  B  ->  R  Fr  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   A.wal 1549   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698   _Vcvv 2948    i^i cin 3311    C_ wss 3312   (/)c0 3620   {csn 3806    Fr wfr 4530   `'ccnv 4869   ran crn 4871   "cima 4873   Fun wfun 5440    Fn wfn 5441   -onto->wfo 5444   -1-1-onto->wf1o 5445   ` cfv 5446    Isom wiso 5447
This theorem is referenced by:  isofr  6054  isofr2  6056  isowe2  6062
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-fr 4533  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455
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