ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cocan1 Unicode version

Theorem cocan1 5681
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 5485 . . . . . 6  |-  ( ( H : A --> B  /\  x  e.  A )  ->  ( ( F  o.  H ) `  x
)  =  ( F `
 ( H `  x ) ) )
213ad2antl2 1144 . . . . 5  |-  ( ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  /\  x  e.  A )  ->  (
( F  o.  H
) `  x )  =  ( F `  ( H `  x ) ) )
3 fvco3 5485 . . . . . 6  |-  ( ( K : A --> B  /\  x  e.  A )  ->  ( ( F  o.  K ) `  x
)  =  ( F `
 ( K `  x ) ) )
433ad2antl3 1145 . . . . 5  |-  ( ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  /\  x  e.  A )  ->  (
( F  o.  K
) `  x )  =  ( F `  ( K `  x ) ) )
52, 4eqeq12d 2152 . . . 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 984 . . . . 5  |-  ( ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  /\  x  e.  A )  ->  F : B -1-1-> C )
7 ffvelrn 5546 . . . . . 6  |-  ( ( H : A --> B  /\  x  e.  A )  ->  ( H `  x
)  e.  B )
873ad2antl2 1144 . . . . 5  |-  ( ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  /\  x  e.  A )  ->  ( H `  x )  e.  B )
9 ffvelrn 5546 . . . . . 6  |-  ( ( K : A --> B  /\  x  e.  A )  ->  ( K `  x
)  e.  B )
1093ad2antl3 1145 . . . . 5  |-  ( ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  /\  x  e.  A )  ->  ( K `  x )  e.  B )
11 f1fveq 5666 . . . . 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 1214 . . . 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 2431 . 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 5323 . . . . . 6  |-  ( F : B -1-1-> C  ->  F : B --> C )
16153ad2ant1 1002 . . . . 5  |-  ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  ->  F : B --> C )
17 ffn 5267 . . . . 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 982 . . . 4  |-  ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  ->  H : A --> B )
20 fnfco 5292 . . . 4  |-  ( ( F  Fn  B  /\  H : A --> B )  ->  ( F  o.  H )  Fn  A
)
2118, 19, 20syl2anc 408 . . 3  |-  ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  ->  ( F  o.  H )  Fn  A
)
22 simp3 983 . . . 4  |-  ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  ->  K : A --> B )
23 fnfco 5292 . . . 4  |-  ( ( F  Fn  B  /\  K : A --> B )  ->  ( F  o.  K )  Fn  A
)
2418, 22, 23syl2anc 408 . . 3  |-  ( ( F : B -1-1-> C  /\  H : A --> B  /\  K : A --> B )  ->  ( F  o.  K )  Fn  A
)
25 eqfnfv 5511 . . 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 408 . 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 5267 . . . 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 5267 . . . 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 5511 . . 3  |-  ( ( H  Fn  A  /\  K  Fn  A )  ->  ( H  =  K  <->  A. x  e.  A  ( H `  x )  =  ( K `  x ) ) )
3228, 30, 31syl2anc 408 . 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 962    = wceq 1331    e. wcel 1480   A.wral 2414    o. ccom 4538    Fn wfn 5113   -->wf 5114   -1-1->wf1 5115   ` cfv 5118
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fv 5126
This theorem is referenced by:  mapen  6733  hashfacen  10572
  Copyright terms: Public domain W3C validator