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Theorem cocan1 5817
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 5612 . . . . . 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 5612 . . . . . 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 2310 . . . 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 5679 . . . . . 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 5679 . . . . . 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 5802 . . . . 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 2572 . 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 5453 . . . . . 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 5405 . . . . 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 5423 . . . 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 5423 . . . 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 5638 . . 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 5405 . . . 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 5405 . . . 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 5638 . . 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 1632    e. wcel 1696   A.wral 2556    o. ccom 4709    Fn wfn 5266   -->wf 5267   -1-1->wf1 5268   ` cfv 5271
This theorem is referenced by:  mapen  7041  mapfien  7415  hashfacen  11408  setcmon  13935  derangenlem  23717  subfacp1lem5  23730  injsurinj  25252
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fv 5279
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