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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 5596 . . . . . 6  |-  ( ( H : A --> B  /\  x  e.  A )  ->  ( ( F  o.  H ) `  x
)  =  ( F `
 ( H `  x ) ) )
213ad2antl2 1118 . . . . 5  |-  ( ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  /\  x  e.  A )  ->  (
( F  o.  H
) `  x )  =  ( F `  ( H `  x ) ) )
3 fvco3 5596 . . . . . 6  |-  ( ( K : A --> B  /\  x  e.  A )  ->  ( ( F  o.  K ) `  x
)  =  ( F `
 ( K `  x ) ) )
433ad2antl3 1119 . . . . 5  |-  ( ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  /\  x  e.  A )  ->  (
( F  o.  K
) `  x )  =  ( F `  ( K `  x ) ) )
52, 4eqeq12d 2297 . . . 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 958 . . . . 5  |-  ( ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  /\  x  e.  A )  ->  F : B -1-1-> C )
7 ffvelrn 5663 . . . . . 6  |-  ( ( H : A --> B  /\  x  e.  A )  ->  ( H `  x
)  e.  B )
873ad2antl2 1118 . . . . 5  |-  ( ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  /\  x  e.  A )  ->  ( H `  x )  e.  B )
9 ffvelrn 5663 . . . . . 6  |-  ( ( K : A --> B  /\  x  e.  A )  ->  ( K `  x
)  e.  B )
1093ad2antl3 1119 . . . . 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 1180 . . . 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 244 . . 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 2559 . 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 5437 . . . . . 6  |-  ( F : B -1-1-> C  ->  F : B --> C )
16153ad2ant1 976 . . . . 5  |-  ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  ->  F : B --> C )
17 ffn 5389 . . . . 5  |-  ( F : B --> C  ->  F  Fn  B )
1816, 17syl 15 . . . 4  |-  ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  ->  F  Fn  B
)
19 simp2 956 . . . 4  |-  ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  ->  H : A --> B )
20 fnfco 5407 . . . 4  |-  ( ( F  Fn  B  /\  H : A --> B )  ->  ( F  o.  H )  Fn  A
)
2118, 19, 20syl2anc 642 . . 3  |-  ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  ->  ( F  o.  H )  Fn  A
)
22 simp3 957 . . . 4  |-  ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  ->  K : A --> B )
23 fnfco 5407 . . . 4  |-  ( ( F  Fn  B  /\  K : A --> B )  ->  ( F  o.  K )  Fn  A
)
2418, 22, 23syl2anc 642 . . 3  |-  ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  ->  ( F  o.  K )  Fn  A
)
25 eqfnfv 5622 . . 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 642 . 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 5389 . . . 4  |-  ( H : A --> B  ->  H  Fn  A )
2819, 27syl 15 . . 3  |-  ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  ->  H  Fn  A
)
29 ffn 5389 . . . 4  |-  ( K : A --> B  ->  K  Fn  A )
3022, 29syl 15 . . 3  |-  ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  ->  K  Fn  A
)
31 eqfnfv 5622 . . 3  |-  ( ( H  Fn  A  /\  K  Fn  A )  ->  ( H  =  K  <->  A. x  e.  A  ( H `  x )  =  ( K `  x ) ) )
3228, 30, 31syl2anc 642 . 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 276 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    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    o. ccom 4693    Fn wfn 5250   -->wf 5251   -1-1->wf1 5252   ` cfv 5255
This theorem is referenced by:  mapen  7025  mapfien  7399  hashfacen  11392  setcmon  13919  derangenlem  23702  subfacp1lem5  23715  injsurinj  25149
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fv 5263
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