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Theorem cocan2 5428
Description: A surjection is right-cancelable. (Contributed by FL, 21-Nov-2011.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
cocan2  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  ( ( H  o.  F )  =  ( K  o.  F )  <->  H  =  K ) )

Proof of Theorem cocan2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fof 5106 . . . . . . 7  |-  ( F : A -onto-> B  ->  F : A --> B )
213ad2ant1 925 . . . . . 6  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  F : A
--> B )
3 fvco3 5244 . . . . . 6  |-  ( ( F : A --> B  /\  y  e.  A )  ->  ( ( H  o.  F ) `  y
)  =  ( H `
 ( F `  y ) ) )
42, 3sylan 267 . . . . 5  |-  ( ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  /\  y  e.  A )  ->  (
( H  o.  F
) `  y )  =  ( H `  ( F `  y ) ) )
5 fvco3 5244 . . . . . 6  |-  ( ( F : A --> B  /\  y  e.  A )  ->  ( ( K  o.  F ) `  y
)  =  ( K `
 ( F `  y ) ) )
62, 5sylan 267 . . . . 5  |-  ( ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  /\  y  e.  A )  ->  (
( K  o.  F
) `  y )  =  ( K `  ( F `  y ) ) )
74, 6eqeq12d 2054 . . . 4  |-  ( ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  /\  y  e.  A )  ->  (
( ( H  o.  F ) `  y
)  =  ( ( K  o.  F ) `
 y )  <->  ( H `  ( F `  y
) )  =  ( K `  ( F `
 y ) ) ) )
87ralbidva 2322 . . 3  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  ( A. y  e.  A  (
( H  o.  F
) `  y )  =  ( ( K  o.  F ) `  y )  <->  A. y  e.  A  ( H `  ( F `  y
) )  =  ( K `  ( F `
 y ) ) ) )
9 fveq2 5178 . . . . . 6  |-  ( ( F `  y )  =  x  ->  ( H `  ( F `  y ) )  =  ( H `  x
) )
10 fveq2 5178 . . . . . 6  |-  ( ( F `  y )  =  x  ->  ( K `  ( F `  y ) )  =  ( K `  x
) )
119, 10eqeq12d 2054 . . . . 5  |-  ( ( F `  y )  =  x  ->  (
( H `  ( F `  y )
)  =  ( K `
 ( F `  y ) )  <->  ( H `  x )  =  ( K `  x ) ) )
1211cbvfo 5425 . . . 4  |-  ( F : A -onto-> B  -> 
( A. y  e.  A  ( H `  ( F `  y ) )  =  ( K `
 ( F `  y ) )  <->  A. x  e.  B  ( H `  x )  =  ( K `  x ) ) )
13123ad2ant1 925 . . 3  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  ( A. y  e.  A  ( H `  ( F `  y ) )  =  ( K `  ( F `  y )
)  <->  A. x  e.  B  ( H `  x )  =  ( K `  x ) ) )
148, 13bitrd 177 . 2  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  ( A. y  e.  A  (
( H  o.  F
) `  y )  =  ( ( K  o.  F ) `  y )  <->  A. x  e.  B  ( H `  x )  =  ( K `  x ) ) )
15 simp2 905 . . . 4  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  H  Fn  B )
16 fnfco 5065 . . . 4  |-  ( ( H  Fn  B  /\  F : A --> B )  ->  ( H  o.  F )  Fn  A
)
1715, 2, 16syl2anc 391 . . 3  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  ( H  o.  F )  Fn  A
)
18 simp3 906 . . . 4  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  K  Fn  B )
19 fnfco 5065 . . . 4  |-  ( ( K  Fn  B  /\  F : A --> B )  ->  ( K  o.  F )  Fn  A
)
2018, 2, 19syl2anc 391 . . 3  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  ( K  o.  F )  Fn  A
)
21 eqfnfv 5265 . . 3  |-  ( ( ( H  o.  F
)  Fn  A  /\  ( K  o.  F
)  Fn  A )  ->  ( ( H  o.  F )  =  ( K  o.  F
)  <->  A. y  e.  A  ( ( H  o.  F ) `  y
)  =  ( ( K  o.  F ) `
 y ) ) )
2217, 20, 21syl2anc 391 . 2  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  ( ( H  o.  F )  =  ( K  o.  F )  <->  A. y  e.  A  ( ( H  o.  F ) `  y )  =  ( ( K  o.  F
) `  y )
) )
23 eqfnfv 5265 . . 3  |-  ( ( H  Fn  B  /\  K  Fn  B )  ->  ( H  =  K  <->  A. x  e.  B  ( H `  x )  =  ( K `  x ) ) )
2415, 18, 23syl2anc 391 . 2  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  ( H  =  K  <->  A. x  e.  B  ( H `  x )  =  ( K `  x ) ) )
2514, 22, 243bitr4d 209 1  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  ( ( H  o.  F )  =  ( K  o.  F )  <->  H  =  K ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    /\ w3a 885    = wceq 1243    e. wcel 1393   A.wral 2306    o. ccom 4349    Fn wfn 4897   -->wf 4898   -onto->wfo 4900   ` cfv 4902
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fo 4908  df-fv 4910
This theorem is referenced by: (None)
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