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Theorem cocan2 5767
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 5420 . . . . . . 7 |- ( F : A -onto-> B -> F : A --> B )
213ad2ant1 1013 . . . . . 6 |- ( ( F : A -onto-> B /\ H Fn B /\ K Fn B ) -> F : A --> B )
3 fvco3 5567 . . . . . 6 |- ( ( F : A --> B /\ y e. A ) -> ( ( H o. F ) ` y ) = ( H ` ( F ` y ) ) )
42, 3sylan 281 . . . . 5 |- ( ( ( F : A -onto-> B /\ H Fn B /\ K Fn B ) /\ y e. A ) -> ( ( H o. F ) ` y ) = ( H ` ( F ` y ) ) )
5 fvco3 5567 . . . . . 6 |- ( ( F : A --> B /\ y e. A ) -> ( ( K o. F ) ` y ) = ( K ` ( F ` y ) ) )
62, 5sylan 281 . . . . 5 |- ( ( ( F : A -onto-> B /\ H Fn B /\ K Fn B ) /\ y e. A ) -> ( ( K o. F ) ` y ) = ( K ` ( F ` y ) ) )
74, 6eqeq12d 2185 . . . 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 2466 . . 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 5496 . . . . . 6 |- ( ( F ` y ) = x -> ( H ` ( F ` y ) ) = ( H ` x ) )
10 fveq2 5496 . . . . . 6 |- ( ( F ` y ) = x -> ( K ` ( F ` y ) ) = ( K ` x ) )
119, 10eqeq12d 2185 . . . . 5 |- ( ( F ` y ) = x -> ( ( H ` ( F ` y ) ) = ( K ` ( F ` y ) ) <-> ( H ` x ) = ( K ` x ) ) )
1211cbvfo 5764 . . . 4 |- ( F : A -onto-> B -> ( A. y e. A ( H ` ( F ` y ) ) = ( K ` ( F ` y ) ) <-> A. x e. B ( H ` x ) = ( K ` x ) ) )
13123ad2ant1 1013 . . 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 187 . 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 993 . . . 4 |- ( ( F : A -onto-> B /\ H Fn B /\ K Fn B ) -> H Fn B )
16 fnfco 5372 . . . 4 |- ( ( H Fn B /\ F : A --> B ) -> ( H o. F ) Fn A )
1715, 2, 16syl2anc 409 . . 3 |- ( ( F : A -onto-> B /\ H Fn B /\ K Fn B ) -> ( H o. F ) Fn A )
18 simp3 994 . . . 4 |- ( ( F : A -onto-> B /\ H Fn B /\ K Fn B ) -> K Fn B )
19 fnfco 5372 . . . 4 |- ( ( K Fn B /\ F : A --> B ) -> ( K o. F ) Fn A )
2018, 2, 19syl2anc 409 . . 3 |- ( ( F : A -onto-> B /\ H Fn B /\ K Fn B ) -> ( K o. F ) Fn A )
21 eqfnfv 5593 . . 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 409 . 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 5593 . . 3 |- ( ( H Fn B /\ K Fn B ) -> ( H = K <-> A. x e. B ( H ` x ) = ( K ` x ) ) )
2415, 18, 23syl2anc 409 . 2 |- ( ( F : A -onto-> B /\ H Fn B /\ K Fn B ) -> ( H = K <-> A. x e. B ( H ` x ) = ( K ` x ) ) )
2514, 22, 243bitr4d 219 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 103   <-> wb 104   /\ w3a 973   = wceq 1348   e. wcel 2141   A.wral 2448   o. ccom 4615   Fn wfn 5193   -->wf 5194   -onto->wfo 5196   ` cfv 5198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fo 5204  df-fv 5206
This theorem is referenced by:  mapen  6824  hashfacen  10771
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