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Theorem cocan1 5755
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 5557 . . . . . 6 |- ( ( H : A --> B /\ x e. A ) -> ( ( F o. H ) ` x ) = ( F ` ( H ` x ) ) )
213ad2antl2 1150 . . . . 5 |- ( ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) /\ x e. A ) -> ( ( F o. H ) ` x ) = ( F ` ( H ` x ) ) )
3 fvco3 5557 . . . . . 6 |- ( ( K : A --> B /\ x e. A ) -> ( ( F o. K ) ` x ) = ( F ` ( K ` x ) ) )
433ad2antl3 1151 . . . . 5 |- ( ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) /\ x e. A ) -> ( ( F o. K ) ` x ) = ( F ` ( K ` x ) ) )
52, 4eqeq12d 2180 . . . 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 990 . . . . 5 |- ( ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) /\ x e. A ) -> F : B -1-1-> C )
7 ffvelrn 5618 . . . . . 6 |- ( ( H : A --> B /\ x e. A ) -> ( H ` x ) e. B )
873ad2antl2 1150 . . . . 5 |- ( ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) /\ x e. A ) -> ( H ` x ) e. B )
9 ffvelrn 5618 . . . . . 6 |- ( ( K : A --> B /\ x e. A ) -> ( K ` x ) e. B )
1093ad2antl3 1151 . . . . 5 |- ( ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) /\ x e. A ) -> ( K ` x ) e. B )
11 f1fveq 5740 . . . . 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 1226 . . . 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 2462 . 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 5393 . . . . . 6 |- ( F : B -1-1-> C -> F : B --> C )
16153ad2ant1 1008 . . . . 5 |- ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) -> F : B --> C )
17 ffn 5337 . . . . 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 988 . . . 4 |- ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) -> H : A --> B )
20 fnfco 5362 . . . 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 989 . . . 4 |- ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) -> K : A --> B )
23 fnfco 5362 . . . 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 5583 . . 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 5337 . . . 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 5337 . . . 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 5583 . . 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 968   = wceq 1343   e. wcel 2136   A.wral 2444   o. ccom 4608   Fn wfn 5183   -->wf 5184   -1-1->wf1 5185   ` cfv 5188
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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187
This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fv 5196
This theorem is referenced by:  mapen  6812  hashfacen  10749
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