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Theorem cocan1 5927
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 5717 . . . . . 6 |- ( ( H : A --> B /\ x e. A ) -> ( ( F o. H ) ` x ) = ( F ` ( H ` x ) ) )
213ad2antl2 1186 . . . . 5 |- ( ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) /\ x e. A ) -> ( ( F o. H ) ` x ) = ( F ` ( H ` x ) ) )
3 fvco3 5717 . . . . . 6 |- ( ( K : A --> B /\ x e. A ) -> ( ( F o. K ) ` x ) = ( F ` ( K ` x ) ) )
433ad2antl3 1187 . . . . 5 |- ( ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) /\ x e. A ) -> ( ( F o. K ) ` x ) = ( F ` ( K ` x ) ) )
52, 4eqeq12d 2246 . . . 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 1026 . . . . 5 |- ( ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) /\ x e. A ) -> F : B -1-1-> C )
7 ffvelcdm 5780 . . . . . 6 |- ( ( H : A --> B /\ x e. A ) -> ( H ` x ) e. B )
873ad2antl2 1186 . . . . 5 |- ( ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) /\ x e. A ) -> ( H ` x ) e. B )
9 ffvelcdm 5780 . . . . . 6 |- ( ( K : A --> B /\ x e. A ) -> ( K ` x ) e. B )
1093ad2antl3 1187 . . . . 5 |- ( ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) /\ x e. A ) -> ( K ` x ) e. B )
11 f1fveq 5912 . . . . 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 1271 . . . 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 2528 . 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 5542 . . . . . 6 |- ( F : B -1-1-> C -> F : B --> C )
16153ad2ant1 1044 . . . . 5 |- ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) -> F : B --> C )
17 ffn 5482 . . . . 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 1024 . . . 4 |- ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) -> H : A --> B )
20 fnfco 5511 . . . 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 1025 . . . 4 |- ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) -> K : A --> B )
23 fnfco 5511 . . . 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 5744 . . 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 5482 . . . 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 5482 . . . 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 5744 . . 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 1004   = wceq 1397   e. wcel 2202   A.wral 2510   o. ccom 4729   Fn wfn 5321   -->wf 5322   -1-1->wf1 5323   ` cfv 5326
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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299
This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fv 5334
This theorem is referenced by:  mapen  7031  hashfacen  11099
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