ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cocan2 Unicode version
Theorem cocan2 5779
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 5430 . . . . . . 7 |- ( F : A -onto-> B -> F : A --> B )
213ad2ant1 1018 . . . . . 6 |- ( ( F : A -onto-> B /\ H Fn B /\ K Fn B ) -> F : A --> B )
3 fvco3 5579 . . . . . 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 5579 . . . . . 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 2190 . . . 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 2471 . . 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 5507 . . . . . 6 |- ( ( F ` y ) = x -> ( H ` ( F ` y ) ) = ( H ` x ) )
10 fveq2 5507 . . . . . 6 |- ( ( F ` y ) = x -> ( K ` ( F ` y ) ) = ( K ` x ) )
119, 10eqeq12d 2190 . . . . 5 |- ( ( F ` y ) = x -> ( ( H ` ( F ` y ) ) = ( K ` ( F ` y ) ) <-> ( H ` x ) = ( K ` x ) ) )
1211cbvfo 5776 . . . 4 |- ( F : A -onto-> B -> ( A. y e. A ( H ` ( F ` y ) ) = ( K ` ( F ` y ) ) <-> A. x e. B ( H ` x ) = ( K ` x ) ) )
13123ad2ant1 1018 . . 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 998 . . . 4 |- ( ( F : A -onto-> B /\ H Fn B /\ K Fn B ) -> H Fn B )
16 fnfco 5382 . . . 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 999 . . . 4 |- ( ( F : A -onto-> B /\ H Fn B /\ K Fn B ) -> K Fn B )
19 fnfco 5382 . . . 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 5605 . . 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 5605 . . 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 978   = wceq 1353   e. wcel 2146   A.wral 2453   o. ccom 4624   Fn wfn 5203   -->wf 5204   -onto->wfo 5206   ` cfv 5208
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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-sbc 2961  df-csb 3056  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-fo 5214  df-fv 5216
This theorem is referenced by:  mapen  6836  hashfacen  10782
  Copyright terms: Public domain W3C validator