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Theorem cocan1 6016
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 5792 . . . . . 6  |-  ( ( H : A --> B  /\  x  e.  A )  ->  ( ( F  o.  H ) `  x
)  =  ( F `
 ( H `  x ) ) )
213ad2antl2 1120 . . . . 5  |-  ( ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  /\  x  e.  A )  ->  (
( F  o.  H
) `  x )  =  ( F `  ( H `  x ) ) )
3 fvco3 5792 . . . . . 6  |-  ( ( K : A --> B  /\  x  e.  A )  ->  ( ( F  o.  K ) `  x
)  =  ( F `
 ( K `  x ) ) )
433ad2antl3 1121 . . . . 5  |-  ( ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  /\  x  e.  A )  ->  (
( F  o.  K
) `  x )  =  ( F `  ( K `  x ) ) )
52, 4eqeq12d 2449 . . . 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 960 . . . . 5  |-  ( ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  /\  x  e.  A )  ->  F : B -1-1-> C )
7 ffvelrn 5860 . . . . . 6  |-  ( ( H : A --> B  /\  x  e.  A )  ->  ( H `  x
)  e.  B )
873ad2antl2 1120 . . . . 5  |-  ( ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  /\  x  e.  A )  ->  ( H `  x )  e.  B )
9 ffvelrn 5860 . . . . . 6  |-  ( ( K : A --> B  /\  x  e.  A )  ->  ( K `  x
)  e.  B )
1093ad2antl3 1121 . . . . 5  |-  ( ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  /\  x  e.  A )  ->  ( K `  x )  e.  B )
11 f1fveq 6000 . . . . 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 1182 . . . 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 245 . . 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 2713 . 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 5631 . . . . . 6  |-  ( F : B -1-1-> C  ->  F : B --> C )
16153ad2ant1 978 . . . . 5  |-  ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  ->  F : B --> C )
17 ffn 5583 . . . . 5  |-  ( F : B --> C  ->  F  Fn  B )
1816, 17syl 16 . . . 4  |-  ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  ->  F  Fn  B
)
19 simp2 958 . . . 4  |-  ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  ->  H : A --> B )
20 fnfco 5601 . . . 4  |-  ( ( F  Fn  B  /\  H : A --> B )  ->  ( F  o.  H )  Fn  A
)
2118, 19, 20syl2anc 643 . . 3  |-  ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  ->  ( F  o.  H )  Fn  A
)
22 simp3 959 . . . 4  |-  ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  ->  K : A --> B )
23 fnfco 5601 . . . 4  |-  ( ( F  Fn  B  /\  K : A --> B )  ->  ( F  o.  K )  Fn  A
)
2418, 22, 23syl2anc 643 . . 3  |-  ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  ->  ( F  o.  K )  Fn  A
)
25 eqfnfv 5819 . . 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 643 . 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 5583 . . . 4  |-  ( H : A --> B  ->  H  Fn  A )
2819, 27syl 16 . . 3  |-  ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  ->  H  Fn  A
)
29 ffn 5583 . . . 4  |-  ( K : A --> B  ->  K  Fn  A )
3022, 29syl 16 . . 3  |-  ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  ->  K  Fn  A
)
31 eqfnfv 5819 . . 3  |-  ( ( H  Fn  A  /\  K  Fn  A )  ->  ( H  =  K  <->  A. x  e.  A  ( H `  x )  =  ( K `  x ) ) )
3228, 30, 31syl2anc 643 . 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 277 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697    o. ccom 4874    Fn wfn 5441   -->wf 5442   -1-1->wf1 5443   ` cfv 5446
This theorem is referenced by:  mapen  7263  mapfien  7645  hashfacen  11695  setcmon  14234  derangenlem  24849  subfacp1lem5  24862
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fv 5454
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