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Theorem cocan1 5458
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 5276 . . . . . 6  |-  ( ( H : A --> B  /\  x  e.  A )  ->  ( ( F  o.  H ) `  x
)  =  ( F `
 ( H `  x ) ) )
213ad2antl2 1102 . . . . 5  |-  ( ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  /\  x  e.  A )  ->  (
( F  o.  H
) `  x )  =  ( F `  ( H `  x ) ) )
3 fvco3 5276 . . . . . 6  |-  ( ( K : A --> B  /\  x  e.  A )  ->  ( ( F  o.  K ) `  x
)  =  ( F `
 ( K `  x ) ) )
433ad2antl3 1103 . . . . 5  |-  ( ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  /\  x  e.  A )  ->  (
( F  o.  K
) `  x )  =  ( F `  ( K `  x ) ) )
52, 4eqeq12d 2096 . . . 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 942 . . . . 5  |-  ( ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  /\  x  e.  A )  ->  F : B -1-1-> C )
7 ffvelrn 5332 . . . . . 6  |-  ( ( H : A --> B  /\  x  e.  A )  ->  ( H `  x
)  e.  B )
873ad2antl2 1102 . . . . 5  |-  ( ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  /\  x  e.  A )  ->  ( H `  x )  e.  B )
9 ffvelrn 5332 . . . . . 6  |-  ( ( K : A --> B  /\  x  e.  A )  ->  ( K `  x
)  e.  B )
1093ad2antl3 1103 . . . . 5  |-  ( ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  /\  x  e.  A )  ->  ( K `  x )  e.  B )
11 f1fveq 5443 . . . . 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 1168 . . . 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 186 . . 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 2365 . 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 5123 . . . . . 6  |-  ( F : B -1-1-> C  ->  F : B --> C )
16153ad2ant1 960 . . . . 5  |-  ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  ->  F : B --> C )
17 ffn 5077 . . . . 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 940 . . . 4  |-  ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  ->  H : A --> B )
20 fnfco 5096 . . . 4  |-  ( ( F  Fn  B  /\  H : A --> B )  ->  ( F  o.  H )  Fn  A
)
2118, 19, 20syl2anc 403 . . 3  |-  ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  ->  ( F  o.  H )  Fn  A
)
22 simp3 941 . . . 4  |-  ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  ->  K : A --> B )
23 fnfco 5096 . . . 4  |-  ( ( F  Fn  B  /\  K : A --> B )  ->  ( F  o.  K )  Fn  A
)
2418, 22, 23syl2anc 403 . . 3  |-  ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  ->  ( F  o.  K )  Fn  A
)
25 eqfnfv 5297 . . 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 403 . 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 5077 . . . 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 5077 . . . 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 5297 . . 3  |-  ( ( H  Fn  A  /\  K  Fn  A )  ->  ( H  =  K  <->  A. x  e.  A  ( H `  x )  =  ( K `  x ) ) )
3228, 30, 31syl2anc 403 . 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 218 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 102    <-> wb 103    /\ w3a 920    = wceq 1285    e. wcel 1434   A.wral 2349    o. ccom 4375    Fn wfn 4927   -->wf 4928   -1-1->wf1 4929   ` cfv 4932
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-csb 2910  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fv 4940
This theorem is referenced by: (None)
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