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Theorem cocan2 5939
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 5568 . . . . . . 7 |- ( F : A -onto-> B -> F : A --> B )
213ad2ant1 1045 . . . . . 6 |- ( ( F : A -onto-> B /\ H Fn B /\ K Fn B ) -> F : A --> B )
3 fvco3 5726 . . . . . 6 |- ( ( F : A --> B /\ y e. A ) -> ( ( H o. F ) ` y ) = ( H ` ( F ` y ) ) )
42, 3sylan 283 . . . . 5 |- ( ( ( F : A -onto-> B /\ H Fn B /\ K Fn B ) /\ y e. A ) -> ( ( H o. F ) ` y ) = ( H ` ( F ` y ) ) )
5 fvco3 5726 . . . . . 6 |- ( ( F : A --> B /\ y e. A ) -> ( ( K o. F ) ` y ) = ( K ` ( F ` y ) ) )
62, 5sylan 283 . . . . 5 |- ( ( ( F : A -onto-> B /\ H Fn B /\ K Fn B ) /\ y e. A ) -> ( ( K o. F ) ` y ) = ( K ` ( F ` y ) ) )
74, 6eqeq12d 2246 . . . 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 2529 . . 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 5648 . . . . . 6 |- ( ( F ` y ) = x -> ( H ` ( F ` y ) ) = ( H ` x ) )
10 fveq2 5648 . . . . . 6 |- ( ( F ` y ) = x -> ( K ` ( F ` y ) ) = ( K ` x ) )
119, 10eqeq12d 2246 . . . . 5 |- ( ( F ` y ) = x -> ( ( H ` ( F ` y ) ) = ( K ` ( F ` y ) ) <-> ( H ` x ) = ( K ` x ) ) )
1211cbvfo 5936 . . . 4 |- ( F : A -onto-> B -> ( A. y e. A ( H ` ( F ` y ) ) = ( K ` ( F ` y ) ) <-> A. x e. B ( H ` x ) = ( K ` x ) ) )
13123ad2ant1 1045 . . 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 188 . 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 1025 . . . 4 |- ( ( F : A -onto-> B /\ H Fn B /\ K Fn B ) -> H Fn B )
16 fnfco 5519 . . . 4 |- ( ( H Fn B /\ F : A --> B ) -> ( H o. F ) Fn A )
1715, 2, 16syl2anc 411 . . 3 |- ( ( F : A -onto-> B /\ H Fn B /\ K Fn B ) -> ( H o. F ) Fn A )
18 simp3 1026 . . . 4 |- ( ( F : A -onto-> B /\ H Fn B /\ K Fn B ) -> K Fn B )
19 fnfco 5519 . . . 4 |- ( ( K Fn B /\ F : A --> B ) -> ( K o. F ) Fn A )
2018, 2, 19syl2anc 411 . . 3 |- ( ( F : A -onto-> B /\ H Fn B /\ K Fn B ) -> ( K o. F ) Fn A )
21 eqfnfv 5753 . . 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 411 . 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 5753 . . 3 |- ( ( H Fn B /\ K Fn B ) -> ( H = K <-> A. x e. B ( H ` x ) = ( K ` x ) ) )
2415, 18, 23syl2anc 411 . 2 |- ( ( F : A -onto-> B /\ H Fn B /\ K Fn B ) -> ( H = K <-> A. x e. B ( H ` x ) = ( K ` x ) ) )
2514, 22, 243bitr4d 220 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 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2202   A.wral 2511   o. ccom 4735   Fn wfn 5328   -->wf 5329   -onto->wfo 5331   ` cfv 5333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305
This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fo 5339  df-fv 5341
This theorem is referenced by:  mapen  7075  hashfacen  11163
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