ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cocan1 Unicode version
Theorem cocan1 5763
Description: An injection is left-cancelable. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
cocan1 |- ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) -> ( ( F o. H ) = ( F o. K ) <-> H = K ) )
Proof of Theorem cocan1
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 fvco3 5565 . . . . . 6 |- ( ( H : A --> B /\ x e. A ) -> ( ( F o. H ) ` x ) = ( F ` ( H ` x ) ) )
213ad2antl2 1155 . . . . 5 |- ( ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) /\ x e. A ) -> ( ( F o. H ) ` x ) = ( F ` ( H ` x ) ) )
3 fvco3 5565 . . . . . 6 |- ( ( K : A --> B /\ x e. A ) -> ( ( F o. K ) ` x ) = ( F ` ( K ` x ) ) )
433ad2antl3 1156 . . . . 5 |- ( ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) /\ x e. A ) -> ( ( F o. K ) ` x ) = ( F ` ( K ` x ) ) )
52, 4eqeq12d 2185 . . . 4 |- ( ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) /\ x e. A ) -> ( ( ( F o. H ) ` x ) = ( ( F o. K ) ` x ) <-> ( F ` ( H ` x ) ) = ( F ` ( K ` x ) ) ) )
6 simpl1 995 . . . . 5 |- ( ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) /\ x e. A ) -> F : B -1-1-> C )
7 ffvelrn 5626 . . . . . 6 |- ( ( H : A --> B /\ x e. A ) -> ( H ` x ) e. B )
873ad2antl2 1155 . . . . 5 |- ( ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) /\ x e. A ) -> ( H ` x ) e. B )
9 ffvelrn 5626 . . . . . 6 |- ( ( K : A --> B /\ x e. A ) -> ( K ` x ) e. B )
1093ad2antl3 1156 . . . . 5 |- ( ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) /\ x e. A ) -> ( K ` x ) e. B )
11 f1fveq 5748 . . . . 5 |- ( ( F : B -1-1-> C /\ ( ( H ` x ) e. B /\ ( K ` x ) e. B ) ) -> ( ( F ` ( H ` x ) ) = ( F ` ( K ` x ) ) <-> ( H ` x ) = ( K ` x ) ) )
126, 8, 10, 11syl12anc 1231 . . . 4 |- ( ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) /\ x e. A ) -> ( ( F ` ( H ` x ) ) = ( F ` ( K ` x ) ) <-> ( H ` x ) = ( K ` x ) ) )
135, 12bitrd 187 . . 3 |- ( ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) /\ x e. A ) -> ( ( ( F o. H ) ` x ) = ( ( F o. K ) ` x ) <-> ( H ` x ) = ( K ` x ) ) )
1413ralbidva 2466 . 2 |- ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) -> ( A. x e. A ( ( F o. H ) ` x ) = ( ( F o. K ) ` x ) <-> A. x e. A ( H ` x ) = ( K ` x ) ) )
15 f1f 5401 . . . . . 6 |- ( F : B -1-1-> C -> F : B --> C )
16153ad2ant1 1013 . . . . 5 |- ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) -> F : B --> C )
17 ffn 5345 . . . . 5 |- ( F : B --> C -> F Fn B )
1816, 17syl 14 . . . 4 |- ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) -> F Fn B )
19 simp2 993 . . . 4 |- ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) -> H : A --> B )
20 fnfco 5370 . . . 4 |- ( ( F Fn B /\ H : A --> B ) -> ( F o. H ) Fn A )
2118, 19, 20syl2anc 409 . . 3 |- ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) -> ( F o. H ) Fn A )
22 simp3 994 . . . 4 |- ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) -> K : A --> B )
23 fnfco 5370 . . . 4 |- ( ( F Fn B /\ K : A --> B ) -> ( F o. K ) Fn A )
2418, 22, 23syl2anc 409 . . 3 |- ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) -> ( F o. K ) Fn A )
25 eqfnfv 5591 . . 3 |- ( ( ( F o. H ) Fn A /\ ( F o. K ) Fn A ) -> ( ( F o. H ) = ( F o. K ) <-> A. x e. A ( ( F o. H ) ` x ) = ( ( F o. K ) ` x ) ) )
2621, 24, 25syl2anc 409 . 2 |- ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) -> ( ( F o. H ) = ( F o. K ) <-> A. x e. A ( ( F o. H ) ` x ) = ( ( F o. K ) ` x ) ) )
27 ffn 5345 . . . 4 |- ( H : A --> B -> H Fn A )
2819, 27syl 14 . . 3 |- ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) -> H Fn A )
29 ffn 5345 . . . 4 |- ( K : A --> B -> K Fn A )
3022, 29syl 14 . . 3 |- ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) -> K Fn A )
31 eqfnfv 5591 . . 3 |- ( ( H Fn A /\ K Fn A ) -> ( H = K <-> A. x e. A ( H ` x ) = ( K ` x ) ) )
3228, 30, 31syl2anc 409 . 2 |- ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) -> ( H = K <-> A. x e. A ( H ` x ) = ( K ` x ) ) )
3314, 26, 323bitr4d 219 1 |- ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) -> ( ( F o. H ) = ( F o. K ) <-> 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 4613   Fn wfn 5191   -->wf 5192   -1-1->wf1 5193   ` cfv 5196
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 4105  ax-pow 4158  ax-pr 4192
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 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fv 5204
This theorem is referenced by:  mapen  6820  hashfacen  10758
  Copyright terms: Public domain W3C validator