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Theorem cocanfo 28623
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 5705 . . . . 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 991 . . . . . . 7  |-  ( ( ( F : A -onto-> B  /\  G  Fn  B  /\  H  Fn  B
)  /\  ( G  o.  F )  =  ( H  o.  F ) )  ->  F : A -onto-> B )
4 fof 5632 . . . . . . 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 5780 . . . . . 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 5780 . . . . . 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 2483 . . . 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 2811 . . 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 5703 . . . . . 6  |-  ( ( F `  y )  =  x  ->  ( G `  ( F `  y ) )  =  ( G `  x
) )
13 fveq2 5703 . . . . . 6  |-  ( ( F `  y )  =  x  ->  ( H `  ( F `  y ) )  =  ( H `  x
) )
1412, 13eqeq12d 2457 . . . . 5  |-  ( ( F `  y )  =  x  ->  (
( G `  ( F `  y )
)  =  ( H `
 ( F `  y ) )  <->  ( G `  x )  =  ( H `  x ) ) )
1514cbvfo 6005 . . . 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 5809 . . . 4  |-  ( ( G  Fn  B  /\  H  Fn  B )  ->  ( G  =  H  <->  A. x  e.  B  ( G `  x )  =  ( H `  x ) ) )
19183adant1 1006 . . 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 965    = wceq 1369    e. wcel 1756   A.wral 2727    o. ccom 4856    Fn wfn 5425   -->wf 5426   -onto->wfo 5428   ` cfv 5430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-fo 5436  df-fv 5438
This theorem is referenced by: (None)
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