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Theorem cocan1 5801
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 5600 . . . . . 6 |- ( ( H : A --> B /\ x e. A ) -> ( ( F o. H ) ` x ) = ( F ` ( H ` x ) ) )
213ad2antl2 1161 . . . . 5 |- ( ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) /\ x e. A ) -> ( ( F o. H ) ` x ) = ( F ` ( H ` x ) ) )
3 fvco3 5600 . . . . . 6 |- ( ( K : A --> B /\ x e. A ) -> ( ( F o. K ) ` x ) = ( F ` ( K ` x ) ) )
433ad2antl3 1162 . . . . 5 |- ( ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) /\ x e. A ) -> ( ( F o. K ) ` x ) = ( F ` ( K ` x ) ) )
52, 4eqeq12d 2202 . . . 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 1001 . . . . 5 |- ( ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) /\ x e. A ) -> F : B -1-1-> C )
7 ffvelcdm 5662 . . . . . 6 |- ( ( H : A --> B /\ x e. A ) -> ( H ` x ) e. B )
873ad2antl2 1161 . . . . 5 |- ( ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) /\ x e. A ) -> ( H ` x ) e. B )
9 ffvelcdm 5662 . . . . . 6 |- ( ( K : A --> B /\ x e. A ) -> ( K ` x ) e. B )
1093ad2antl3 1162 . . . . 5 |- ( ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) /\ x e. A ) -> ( K ` x ) e. B )
11 f1fveq 5786 . . . . 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 1246 . . . 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 2483 . 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 5433 . . . . . 6 |- ( F : B -1-1-> C -> F : B --> C )
16153ad2ant1 1019 . . . . 5 |- ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) -> F : B --> C )
17 ffn 5377 . . . . 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 999 . . . 4 |- ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) -> H : A --> B )
20 fnfco 5402 . . . 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 1000 . . . 4 |- ( ( F : B -1-1-> C /\ H : A --> B /\ K : A --> B ) -> K : A --> B )
23 fnfco 5402 . . . 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 5626 . . 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 5377 . . . 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 5377 . . . 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 5626 . . 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 979   = wceq 1363   e. wcel 2158   A.wral 2465   o. ccom 4642   Fn wfn 5223   -->wf 5224   -1-1->wf1 5225   ` cfv 5228
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-sbc 2975  df-csb 3070  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fv 5236
This theorem is referenced by:  mapen  6860  hashfacen  10830
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