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Theorem cocanfo 26374
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 731 . . . . . 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 5527 . . . . 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 958 . . . . . . 7  |-  ( ( ( F : A -onto-> B  /\  G  Fn  B  /\  H  Fn  B
)  /\  ( G  o.  F )  =  ( H  o.  F ) )  ->  F : A -onto-> B )
4 fof 5451 . . . . . . 7  |-  ( F : A -onto-> B  ->  F : A --> B )
53, 4syl 15 . . . . . 6  |-  ( ( ( F : A -onto-> B  /\  G  Fn  B  /\  H  Fn  B
)  /\  ( G  o.  F )  =  ( H  o.  F ) )  ->  F : A
--> B )
6 fvco3 5596 . . . . . 6  |-  ( ( F : A --> B  /\  y  e.  A )  ->  ( ( G  o.  F ) `  y
)  =  ( G `
 ( F `  y ) ) )
75, 6sylan 457 . . . . 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 5596 . . . . . 6  |-  ( ( F : A --> B  /\  y  e.  A )  ->  ( ( H  o.  F ) `  y
)  =  ( H `
 ( F `  y ) ) )
95, 8sylan 457 . . . . 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 2323 . . . 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 2626 . . 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 5525 . . . . . 6  |-  ( ( F `  y )  =  x  ->  ( G `  ( F `  y ) )  =  ( G `  x
) )
13 fveq2 5525 . . . . . 6  |-  ( ( F `  y )  =  x  ->  ( H `  ( F `  y ) )  =  ( H `  x
) )
1412, 13eqeq12d 2297 . . . . 5  |-  ( ( F `  y )  =  x  ->  (
( G `  ( F `  y )
)  =  ( H `
 ( F `  y ) )  <->  ( G `  x )  =  ( H `  x ) ) )
1514cbvfo 5799 . . . 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 15 . . 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 201 . 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 5622 . . . 4  |-  ( ( G  Fn  B  /\  H  Fn  B )  ->  ( G  =  H  <->  A. x  e.  B  ( G `  x )  =  ( H `  x ) ) )
19183adant1 973 . . 3  |-  ( ( F : A -onto-> B  /\  G  Fn  B  /\  H  Fn  B
)  ->  ( G  =  H  <->  A. x  e.  B  ( G `  x )  =  ( H `  x ) ) )
2019adantr 451 . 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 223 1  |-  ( ( ( F : A -onto-> B  /\  G  Fn  B  /\  H  Fn  B
)  /\  ( G  o.  F )  =  ( H  o.  F ) )  ->  G  =  H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    o. ccom 4693    Fn wfn 5250   -->wf 5251   -onto->wfo 5253   ` cfv 5255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263
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