ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cocan2 Unicode version
Theorem cocan2 5810
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 5457 . . . . . . 7 |- ( F : A -onto-> B -> F : A --> B )
213ad2ant1 1020 . . . . . 6 |- ( ( F : A -onto-> B /\ H Fn B /\ K Fn B ) -> F : A --> B )
3 fvco3 5608 . . . . . 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 5608 . . . . . 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 2204 . . . 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 2486 . . 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 5534 . . . . . 6 |- ( ( F ` y ) = x -> ( H ` ( F ` y ) ) = ( H ` x ) )
10 fveq2 5534 . . . . . 6 |- ( ( F ` y ) = x -> ( K ` ( F ` y ) ) = ( K ` x ) )
119, 10eqeq12d 2204 . . . . 5 |- ( ( F ` y ) = x -> ( ( H ` ( F ` y ) ) = ( K ` ( F ` y ) ) <-> ( H ` x ) = ( K ` x ) ) )
1211cbvfo 5807 . . . 4 |- ( F : A -onto-> B -> ( A. y e. A ( H ` ( F ` y ) ) = ( K ` ( F ` y ) ) <-> A. x e. B ( H ` x ) = ( K ` x ) ) )
13123ad2ant1 1020 . . 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 1000 . . . 4 |- ( ( F : A -onto-> B /\ H Fn B /\ K Fn B ) -> H Fn B )
16 fnfco 5409 . . . 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 1001 . . . 4 |- ( ( F : A -onto-> B /\ H Fn B /\ K Fn B ) -> K Fn B )
19 fnfco 5409 . . . 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 5634 . . 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 5634 . . 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 980   = wceq 1364   e. wcel 2160   A.wral 2468   o. ccom 4648   Fn wfn 5230   -->wf 5231   -onto->wfo 5233   ` cfv 5235
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fo 5241  df-fv 5243
This theorem is referenced by:  mapen  6875  hashfacen  10851
  Copyright terms: Public domain W3C validator