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Theorem cocan1 5837
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 5635 . . . . . 6 |- ( ( H : A --> B /\ x e. A ) -> ( ( F o. H ) ` x ) = ( F ` ( H ` x ) ) )
213ad2antl2 1162 . . . . 5 |- ( ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) /\ x e. A ) -> ( ( F o. H ) ` x ) = ( F ` ( H ` x ) ) )
3 fvco3 5635 . . . . . 6 |- ( ( K : A --> B /\ x e. A ) -> ( ( F o. K ) ` x ) = ( F ` ( K ` x ) ) )
433ad2antl3 1163 . . . . 5 |- ( ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) /\ x e. A ) -> ( ( F o. K ) ` x ) = ( F ` ( K ` x ) ) )
52, 4eqeq12d 2211 . . . 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 1002 . . . . 5 |- ( ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) /\ x e. A ) -> F : B -1-1-> C )
7 ffvelcdm 5698 . . . . . 6 |- ( ( H : A --> B /\ x e. A ) -> ( H ` x ) e. B )
873ad2antl2 1162 . . . . 5 |- ( ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) /\ x e. A ) -> ( H ` x ) e. B )
9 ffvelcdm 5698 . . . . . 6 |- ( ( K : A --> B /\ x e. A ) -> ( K ` x ) e. B )
1093ad2antl3 1163 . . . . 5 |- ( ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) /\ x e. A ) -> ( K ` x ) e. B )
11 f1fveq 5822 . . . . 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 1247 . . . 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 188 . . 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 2493 . 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 5466 . . . . . 6 |- ( F : B -1-1-> C -> F : B --> C )
16153ad2ant1 1020 . . . . 5 |- ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) -> F : B --> C )
17 ffn 5410 . . . . 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 1000 . . . 4 |- ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) -> H : A --> B )
20 fnfco 5435 . . . 4 |- ( ( F Fn B /\ H : A --> B ) -> ( F o. H ) Fn A )
2118, 19, 20syl2anc 411 . . 3 |- ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) -> ( F o. H ) Fn A )
22 simp3 1001 . . . 4 |- ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) -> K : A --> B )
23 fnfco 5435 . . . 4 |- ( ( F Fn B /\ K : A --> B ) -> ( F o. K ) Fn A )
2418, 22, 23syl2anc 411 . . 3 |- ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) -> ( F o. K ) Fn A )
25 eqfnfv 5662 . . 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 411 . 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 5410 . . . 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 5410 . . . 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 5662 . . 3 |- ( ( H Fn A /\ K Fn A ) -> ( H = K <-> A. x e. A ( H ` x ) = ( K ` x ) ) )
3228, 30, 31syl2anc 411 . 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 220 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 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2167   A.wral 2475   o. ccom 4668   Fn wfn 5254   -->wf 5255   -1-1->wf1 5256   ` cfv 5259
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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fv 5267
This theorem is referenced by:  mapen  6916  hashfacen  10945
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