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Theorem cocanfo 26317
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 732 . . . . . 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 5697 . . . . 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 960 . . . . . . 7  |-  ( ( ( F : A -onto-> B  /\  G  Fn  B  /\  H  Fn  B
)  /\  ( G  o.  F )  =  ( H  o.  F ) )  ->  F : A -onto-> B )
4 fof 5620 . . . . . . 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 5767 . . . . . 6  |-  ( ( F : A --> B  /\  y  e.  A )  ->  ( ( G  o.  F ) `  y
)  =  ( G `
 ( F `  y ) ) )
75, 6sylan 458 . . . . 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 5767 . . . . . 6  |-  ( ( F : A --> B  /\  y  e.  A )  ->  ( ( H  o.  F ) `  y
)  =  ( H `
 ( F `  y ) ) )
95, 8sylan 458 . . . . 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 2452 . . . 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 2757 . . 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 5695 . . . . . 6  |-  ( ( F `  y )  =  x  ->  ( G `  ( F `  y ) )  =  ( G `  x
) )
13 fveq2 5695 . . . . . 6  |-  ( ( F `  y )  =  x  ->  ( H `  ( F `  y ) )  =  ( H `  x
) )
1412, 13eqeq12d 2426 . . . . 5  |-  ( ( F `  y )  =  x  ->  (
( G `  ( F `  y )
)  =  ( H `
 ( F `  y ) )  <->  ( G `  x )  =  ( H `  x ) ) )
1514cbvfo 5989 . . . 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 202 . 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 5794 . . . 4  |-  ( ( G  Fn  B  /\  H  Fn  B )  ->  ( G  =  H  <->  A. x  e.  B  ( G `  x )  =  ( H `  x ) ) )
19183adant1 975 . . 3  |-  ( ( F : A -onto-> B  /\  G  Fn  B  /\  H  Fn  B
)  ->  ( G  =  H  <->  A. x  e.  B  ( G `  x )  =  ( H `  x ) ) )
2019adantr 452 . 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 224 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2674    o. ccom 4849    Fn wfn 5416   -->wf 5417   -onto->wfo 5419   ` cfv 5421
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-fo 5427  df-fv 5429
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