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Theorem cocanfo 30038
Description: Cancellation of a surjective function from the right side of a composition. (Contributed by Jeff Madsen, 1-Jun-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
Assertion
Ref Expression
cocanfo  |-  ( ( ( F : A -onto-> B  /\  G  Fn  B  /\  H  Fn  B
)  /\  ( G  o.  F )  =  ( H  o.  F ) )  ->  G  =  H )

Proof of Theorem cocanfo
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplr 754 . . . . . 6  |-  ( ( ( ( F : A -onto-> B  /\  G  Fn  B  /\  H  Fn  B
)  /\  ( G  o.  F )  =  ( H  o.  F ) )  /\  y  e.  A )  ->  ( G  o.  F )  =  ( H  o.  F ) )
21fveq1d 5868 . . . . 5  |-  ( ( ( ( F : A -onto-> B  /\  G  Fn  B  /\  H  Fn  B
)  /\  ( G  o.  F )  =  ( H  o.  F ) )  /\  y  e.  A )  ->  (
( G  o.  F
) `  y )  =  ( ( H  o.  F ) `  y ) )
3 simpl1 999 . . . . . . 7  |-  ( ( ( F : A -onto-> B  /\  G  Fn  B  /\  H  Fn  B
)  /\  ( G  o.  F )  =  ( H  o.  F ) )  ->  F : A -onto-> B )
4 fof 5795 . . . . . . 7  |-  ( F : A -onto-> B  ->  F : A --> B )
53, 4syl 16 . . . . . 6  |-  ( ( ( F : A -onto-> B  /\  G  Fn  B  /\  H  Fn  B
)  /\  ( G  o.  F )  =  ( H  o.  F ) )  ->  F : A
--> B )
6 fvco3 5945 . . . . . 6  |-  ( ( F : A --> B  /\  y  e.  A )  ->  ( ( G  o.  F ) `  y
)  =  ( G `
 ( F `  y ) ) )
75, 6sylan 471 . . . . 5  |-  ( ( ( ( F : A -onto-> B  /\  G  Fn  B  /\  H  Fn  B
)  /\  ( G  o.  F )  =  ( H  o.  F ) )  /\  y  e.  A )  ->  (
( G  o.  F
) `  y )  =  ( G `  ( F `  y ) ) )
8 fvco3 5945 . . . . . 6  |-  ( ( F : A --> B  /\  y  e.  A )  ->  ( ( H  o.  F ) `  y
)  =  ( H `
 ( F `  y ) ) )
95, 8sylan 471 . . . . 5  |-  ( ( ( ( F : A -onto-> B  /\  G  Fn  B  /\  H  Fn  B
)  /\  ( G  o.  F )  =  ( H  o.  F ) )  /\  y  e.  A )  ->  (
( H  o.  F
) `  y )  =  ( H `  ( F `  y ) ) )
102, 7, 93eqtr3d 2516 . . . 4  |-  ( ( ( ( F : A -onto-> B  /\  G  Fn  B  /\  H  Fn  B
)  /\  ( G  o.  F )  =  ( H  o.  F ) )  /\  y  e.  A )  ->  ( G `  ( F `  y ) )  =  ( H `  ( F `  y )
) )
1110ralrimiva 2878 . . 3  |-  ( ( ( F : A -onto-> B  /\  G  Fn  B  /\  H  Fn  B
)  /\  ( G  o.  F )  =  ( H  o.  F ) )  ->  A. y  e.  A  ( G `  ( F `  y
) )  =  ( H `  ( F `
 y ) ) )
12 fveq2 5866 . . . . . 6  |-  ( ( F `  y )  =  x  ->  ( G `  ( F `  y ) )  =  ( G `  x
) )
13 fveq2 5866 . . . . . 6  |-  ( ( F `  y )  =  x  ->  ( H `  ( F `  y ) )  =  ( H `  x
) )
1412, 13eqeq12d 2489 . . . . 5  |-  ( ( F `  y )  =  x  ->  (
( G `  ( F `  y )
)  =  ( H `
 ( F `  y ) )  <->  ( G `  x )  =  ( H `  x ) ) )
1514cbvfo 6181 . . . 4  |-  ( F : A -onto-> B  -> 
( A. y  e.  A  ( G `  ( F `  y ) )  =  ( H `
 ( F `  y ) )  <->  A. x  e.  B  ( G `  x )  =  ( H `  x ) ) )
163, 15syl 16 . . 3  |-  ( ( ( F : A -onto-> B  /\  G  Fn  B  /\  H  Fn  B
)  /\  ( G  o.  F )  =  ( H  o.  F ) )  ->  ( A. y  e.  A  ( G `  ( F `  y ) )  =  ( H `  ( F `  y )
)  <->  A. x  e.  B  ( G `  x )  =  ( H `  x ) ) )
1711, 16mpbid 210 . 2  |-  ( ( ( F : A -onto-> B  /\  G  Fn  B  /\  H  Fn  B
)  /\  ( G  o.  F )  =  ( H  o.  F ) )  ->  A. x  e.  B  ( G `  x )  =  ( H `  x ) )
18 eqfnfv 5976 . . . 4  |-  ( ( G  Fn  B  /\  H  Fn  B )  ->  ( G  =  H  <->  A. x  e.  B  ( G `  x )  =  ( H `  x ) ) )
19183adant1 1014 . . 3  |-  ( ( F : A -onto-> B  /\  G  Fn  B  /\  H  Fn  B
)  ->  ( G  =  H  <->  A. x  e.  B  ( G `  x )  =  ( H `  x ) ) )
2019adantr 465 . 2  |-  ( ( ( F : A -onto-> B  /\  G  Fn  B  /\  H  Fn  B
)  /\  ( G  o.  F )  =  ( H  o.  F ) )  ->  ( G  =  H  <->  A. x  e.  B  ( G `  x )  =  ( H `  x ) ) )
2117, 20mpbird 232 1  |-  ( ( ( F : A -onto-> B  /\  G  Fn  B  /\  H  Fn  B
)  /\  ( G  o.  F )  =  ( H  o.  F ) )  ->  G  =  H )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814    o. ccom 5003    Fn wfn 5583   -->wf 5584   -onto->wfo 5586   ` cfv 5588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fo 5594  df-fv 5596
This theorem is referenced by: (None)
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