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Theorem fvcofneq 5953
Description: The values of two function compositions are equal if the values of the composed functions are pairwise equal. (Contributed by AV, 26-Jan-2019.)
Assertion
Ref Expression
fvcofneq  |-  ( ( G  Fn  A  /\  K  Fn  B )  ->  ( ( X  e.  ( A  i^i  B
)  /\  ( G `  X )  =  ( K `  X )  /\  A. x  e.  ( ran  G  i^i  ran 
K ) ( F `
 x )  =  ( H `  x
) )  ->  (
( F  o.  G
) `  X )  =  ( ( H  o.  K ) `  X ) ) )
Distinct variable groups:    x, F    x, G    x, H    x, K    x, X
Allowed substitution hints:    A( x)    B( x)

Proof of Theorem fvcofneq
StepHypRef Expression
1 simpl 457 . . . 4  |-  ( ( G  Fn  A  /\  K  Fn  B )  ->  G  Fn  A )
2 elin 3640 . . . . . 6  |-  ( X  e.  ( A  i^i  B )  <->  ( X  e.  A  /\  X  e.  B ) )
3 simpl 457 . . . . . 6  |-  ( ( X  e.  A  /\  X  e.  B )  ->  X  e.  A )
42, 3sylbi 195 . . . . 5  |-  ( X  e.  ( A  i^i  B )  ->  X  e.  A )
543ad2ant1 1009 . . . 4  |-  ( ( X  e.  ( A  i^i  B )  /\  ( G `  X )  =  ( K `  X )  /\  A. x  e.  ( ran  G  i^i  ran  K )
( F `  x
)  =  ( H `
 x ) )  ->  X  e.  A
)
6 fvco2 5868 . . . 4  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( ( F  o.  G ) `  X
)  =  ( F `
 ( G `  X ) ) )
71, 5, 6syl2an 477 . . 3  |-  ( ( ( G  Fn  A  /\  K  Fn  B
)  /\  ( X  e.  ( A  i^i  B
)  /\  ( G `  X )  =  ( K `  X )  /\  A. x  e.  ( ran  G  i^i  ran 
K ) ( F `
 x )  =  ( H `  x
) ) )  -> 
( ( F  o.  G ) `  X
)  =  ( F `
 ( G `  X ) ) )
8 simpr 461 . . . . 5  |-  ( ( G  Fn  A  /\  K  Fn  B )  ->  K  Fn  B )
9 simpr 461 . . . . . . 7  |-  ( ( X  e.  A  /\  X  e.  B )  ->  X  e.  B )
102, 9sylbi 195 . . . . . 6  |-  ( X  e.  ( A  i^i  B )  ->  X  e.  B )
11103ad2ant1 1009 . . . . 5  |-  ( ( X  e.  ( A  i^i  B )  /\  ( G `  X )  =  ( K `  X )  /\  A. x  e.  ( ran  G  i^i  ran  K )
( F `  x
)  =  ( H `
 x ) )  ->  X  e.  B
)
12 fvco2 5868 . . . . 5  |-  ( ( K  Fn  B  /\  X  e.  B )  ->  ( ( H  o.  K ) `  X
)  =  ( H `
 ( K `  X ) ) )
138, 11, 12syl2an 477 . . . 4  |-  ( ( ( G  Fn  A  /\  K  Fn  B
)  /\  ( X  e.  ( A  i^i  B
)  /\  ( G `  X )  =  ( K `  X )  /\  A. x  e.  ( ran  G  i^i  ran 
K ) ( F `
 x )  =  ( H `  x
) ) )  -> 
( ( H  o.  K ) `  X
)  =  ( H `
 ( K `  X ) ) )
14 fveq2 5792 . . . . . . 7  |-  ( ( K `  X )  =  ( G `  X )  ->  ( H `  ( K `  X ) )  =  ( H `  ( G `  X )
) )
1514eqcoms 2463 . . . . . 6  |-  ( ( G `  X )  =  ( K `  X )  ->  ( H `  ( K `  X ) )  =  ( H `  ( G `  X )
) )
16153ad2ant2 1010 . . . . 5  |-  ( ( X  e.  ( A  i^i  B )  /\  ( G `  X )  =  ( K `  X )  /\  A. x  e.  ( ran  G  i^i  ran  K )
( F `  x
)  =  ( H `
 x ) )  ->  ( H `  ( K `  X ) )  =  ( H `
 ( G `  X ) ) )
1716adantl 466 . . . 4  |-  ( ( ( G  Fn  A  /\  K  Fn  B
)  /\  ( X  e.  ( A  i^i  B
)  /\  ( G `  X )  =  ( K `  X )  /\  A. x  e.  ( ran  G  i^i  ran 
K ) ( F `
 x )  =  ( H `  x
) ) )  -> 
( H `  ( K `  X )
)  =  ( H `
 ( G `  X ) ) )
18 id 22 . . . . . . . . . . . 12  |-  ( G  Fn  A  ->  G  Fn  A )
19 fnfvelrn 5942 . . . . . . . . . . . 12  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( G `  X
)  e.  ran  G
)
2018, 4, 19syl2anr 478 . . . . . . . . . . 11  |-  ( ( X  e.  ( A  i^i  B )  /\  G  Fn  A )  ->  ( G `  X
)  e.  ran  G
)
2120ex 434 . . . . . . . . . 10  |-  ( X  e.  ( A  i^i  B )  ->  ( G  Fn  A  ->  ( G `
 X )  e. 
ran  G ) )
22 id 22 . . . . . . . . . . . 12  |-  ( K  Fn  B  ->  K  Fn  B )
23 fnfvelrn 5942 . . . . . . . . . . . 12  |-  ( ( K  Fn  B  /\  X  e.  B )  ->  ( K `  X
)  e.  ran  K
)
2422, 10, 23syl2anr 478 . . . . . . . . . . 11  |-  ( ( X  e.  ( A  i^i  B )  /\  K  Fn  B )  ->  ( K `  X
)  e.  ran  K
)
2524ex 434 . . . . . . . . . 10  |-  ( X  e.  ( A  i^i  B )  ->  ( K  Fn  B  ->  ( K `
 X )  e. 
ran  K ) )
2621, 25anim12d 563 . . . . . . . . 9  |-  ( X  e.  ( A  i^i  B )  ->  ( ( G  Fn  A  /\  K  Fn  B )  ->  ( ( G `  X )  e.  ran  G  /\  ( K `  X )  e.  ran  K ) ) )
27 eleq1 2523 . . . . . . . . . . . 12  |-  ( ( K `  X )  =  ( G `  X )  ->  (
( K `  X
)  e.  ran  K  <->  ( G `  X )  e.  ran  K ) )
2827eqcoms 2463 . . . . . . . . . . 11  |-  ( ( G `  X )  =  ( K `  X )  ->  (
( K `  X
)  e.  ran  K  <->  ( G `  X )  e.  ran  K ) )
2928anbi2d 703 . . . . . . . . . 10  |-  ( ( G `  X )  =  ( K `  X )  ->  (
( ( G `  X )  e.  ran  G  /\  ( K `  X )  e.  ran  K )  <->  ( ( G `
 X )  e. 
ran  G  /\  ( G `  X )  e.  ran  K ) ) )
30 elin 3640 . . . . . . . . . . 11  |-  ( ( G `  X )  e.  ( ran  G  i^i  ran  K )  <->  ( ( G `  X )  e.  ran  G  /\  ( G `  X )  e.  ran  K ) )
3130biimpri 206 . . . . . . . . . 10  |-  ( ( ( G `  X
)  e.  ran  G  /\  ( G `  X
)  e.  ran  K
)  ->  ( G `  X )  e.  ( ran  G  i^i  ran  K ) )
3229, 31syl6bi 228 . . . . . . . . 9  |-  ( ( G `  X )  =  ( K `  X )  ->  (
( ( G `  X )  e.  ran  G  /\  ( K `  X )  e.  ran  K )  ->  ( G `  X )  e.  ( ran  G  i^i  ran  K ) ) )
3326, 32sylan9 657 . . . . . . . 8  |-  ( ( X  e.  ( A  i^i  B )  /\  ( G `  X )  =  ( K `  X ) )  -> 
( ( G  Fn  A  /\  K  Fn  B
)  ->  ( G `  X )  e.  ( ran  G  i^i  ran  K ) ) )
34 fveq2 5792 . . . . . . . . . . . 12  |-  ( x  =  ( G `  X )  ->  ( F `  x )  =  ( F `  ( G `  X ) ) )
35 fveq2 5792 . . . . . . . . . . . 12  |-  ( x  =  ( G `  X )  ->  ( H `  x )  =  ( H `  ( G `  X ) ) )
3634, 35eqeq12d 2473 . . . . . . . . . . 11  |-  ( x  =  ( G `  X )  ->  (
( F `  x
)  =  ( H `
 x )  <->  ( F `  ( G `  X
) )  =  ( H `  ( G `
 X ) ) ) )
3736rspcva 3170 . . . . . . . . . 10  |-  ( ( ( G `  X
)  e.  ( ran 
G  i^i  ran  K )  /\  A. x  e.  ( ran  G  i^i  ran 
K ) ( F `
 x )  =  ( H `  x
) )  ->  ( F `  ( G `  X ) )  =  ( H `  ( G `  X )
) )
3837eqcomd 2459 . . . . . . . . 9  |-  ( ( ( G `  X
)  e.  ( ran 
G  i^i  ran  K )  /\  A. x  e.  ( ran  G  i^i  ran 
K ) ( F `
 x )  =  ( H `  x
) )  ->  ( H `  ( G `  X ) )  =  ( F `  ( G `  X )
) )
3938ex 434 . . . . . . . 8  |-  ( ( G `  X )  e.  ( ran  G  i^i  ran  K )  -> 
( A. x  e.  ( ran  G  i^i  ran 
K ) ( F `
 x )  =  ( H `  x
)  ->  ( H `  ( G `  X
) )  =  ( F `  ( G `
 X ) ) ) )
4033, 39syl6 33 . . . . . . 7  |-  ( ( X  e.  ( A  i^i  B )  /\  ( G `  X )  =  ( K `  X ) )  -> 
( ( G  Fn  A  /\  K  Fn  B
)  ->  ( A. x  e.  ( ran  G  i^i  ran  K )
( F `  x
)  =  ( H `
 x )  -> 
( H `  ( G `  X )
)  =  ( F `
 ( G `  X ) ) ) ) )
4140com23 78 . . . . . 6  |-  ( ( X  e.  ( A  i^i  B )  /\  ( G `  X )  =  ( K `  X ) )  -> 
( A. x  e.  ( ran  G  i^i  ran 
K ) ( F `
 x )  =  ( H `  x
)  ->  ( ( G  Fn  A  /\  K  Fn  B )  ->  ( H `  ( G `  X )
)  =  ( F `
 ( G `  X ) ) ) ) )
42413impia 1185 . . . . 5  |-  ( ( X  e.  ( A  i^i  B )  /\  ( G `  X )  =  ( K `  X )  /\  A. x  e.  ( ran  G  i^i  ran  K )
( F `  x
)  =  ( H `
 x ) )  ->  ( ( G  Fn  A  /\  K  Fn  B )  ->  ( H `  ( G `  X ) )  =  ( F `  ( G `  X )
) ) )
4342impcom 430 . . . 4  |-  ( ( ( G  Fn  A  /\  K  Fn  B
)  /\  ( X  e.  ( A  i^i  B
)  /\  ( G `  X )  =  ( K `  X )  /\  A. x  e.  ( ran  G  i^i  ran 
K ) ( F `
 x )  =  ( H `  x
) ) )  -> 
( H `  ( G `  X )
)  =  ( F `
 ( G `  X ) ) )
4413, 17, 433eqtrrd 2497 . . 3  |-  ( ( ( G  Fn  A  /\  K  Fn  B
)  /\  ( X  e.  ( A  i^i  B
)  /\  ( G `  X )  =  ( K `  X )  /\  A. x  e.  ( ran  G  i^i  ran 
K ) ( F `
 x )  =  ( H `  x
) ) )  -> 
( F `  ( G `  X )
)  =  ( ( H  o.  K ) `
 X ) )
457, 44eqtrd 2492 . 2  |-  ( ( ( G  Fn  A  /\  K  Fn  B
)  /\  ( X  e.  ( A  i^i  B
)  /\  ( G `  X )  =  ( K `  X )  /\  A. x  e.  ( ran  G  i^i  ran 
K ) ( F `
 x )  =  ( H `  x
) ) )  -> 
( ( F  o.  G ) `  X
)  =  ( ( H  o.  K ) `
 X ) )
4645ex 434 1  |-  ( ( G  Fn  A  /\  K  Fn  B )  ->  ( ( X  e.  ( A  i^i  B
)  /\  ( G `  X )  =  ( K `  X )  /\  A. x  e.  ( ran  G  i^i  ran 
K ) ( F `
 x )  =  ( H `  x
) )  ->  (
( F  o.  G
) `  X )  =  ( ( H  o.  K ) `  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2795    i^i cin 3428   ran crn 4942    o. ccom 4945    Fn wfn 5514   ` cfv 5519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-fv 5527
This theorem is referenced by:  fvcosymgeq  16045
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