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Theorem cocan2 5818
Description: A surjection is right-cancelable. (Contributed by FL, 21-Nov-2011.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
cocan2  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  ( ( H  o.  F )  =  ( K  o.  F )  <->  H  =  K ) )

Proof of Theorem cocan2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fof 5467 . . . . . . 7  |-  ( F : A -onto-> B  ->  F : A --> B )
213ad2ant1 976 . . . . . 6  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  F : A
--> B )
3 fvco3 5612 . . . . . 6  |-  ( ( F : A --> B  /\  y  e.  A )  ->  ( ( H  o.  F ) `  y
)  =  ( H `
 ( F `  y ) ) )
42, 3sylan 457 . . . . 5  |-  ( ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  /\  y  e.  A )  ->  (
( H  o.  F
) `  y )  =  ( H `  ( F `  y ) ) )
5 fvco3 5612 . . . . . 6  |-  ( ( F : A --> B  /\  y  e.  A )  ->  ( ( K  o.  F ) `  y
)  =  ( K `
 ( F `  y ) ) )
62, 5sylan 457 . . . . 5  |-  ( ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  /\  y  e.  A )  ->  (
( K  o.  F
) `  y )  =  ( K `  ( F `  y ) ) )
74, 6eqeq12d 2310 . . . 4  |-  ( ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  /\  y  e.  A )  ->  (
( ( H  o.  F ) `  y
)  =  ( ( K  o.  F ) `
 y )  <->  ( H `  ( F `  y
) )  =  ( K `  ( F `
 y ) ) ) )
87ralbidva 2572 . . 3  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  ( A. y  e.  A  (
( H  o.  F
) `  y )  =  ( ( K  o.  F ) `  y )  <->  A. y  e.  A  ( H `  ( F `  y
) )  =  ( K `  ( F `
 y ) ) ) )
9 fveq2 5541 . . . . . 6  |-  ( ( F `  y )  =  x  ->  ( H `  ( F `  y ) )  =  ( H `  x
) )
10 fveq2 5541 . . . . . 6  |-  ( ( F `  y )  =  x  ->  ( K `  ( F `  y ) )  =  ( K `  x
) )
119, 10eqeq12d 2310 . . . . 5  |-  ( ( F `  y )  =  x  ->  (
( H `  ( F `  y )
)  =  ( K `
 ( F `  y ) )  <->  ( H `  x )  =  ( K `  x ) ) )
1211cbvfo 5815 . . . 4  |-  ( F : A -onto-> B  -> 
( A. y  e.  A  ( H `  ( F `  y ) )  =  ( K `
 ( F `  y ) )  <->  A. x  e.  B  ( H `  x )  =  ( K `  x ) ) )
13123ad2ant1 976 . . 3  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  ( A. y  e.  A  ( H `  ( F `  y ) )  =  ( K `  ( F `  y )
)  <->  A. x  e.  B  ( H `  x )  =  ( K `  x ) ) )
148, 13bitrd 244 . 2  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  ( A. y  e.  A  (
( H  o.  F
) `  y )  =  ( ( K  o.  F ) `  y )  <->  A. x  e.  B  ( H `  x )  =  ( K `  x ) ) )
15 simp2 956 . . . 4  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  H  Fn  B )
16 fnfco 5423 . . . 4  |-  ( ( H  Fn  B  /\  F : A --> B )  ->  ( H  o.  F )  Fn  A
)
1715, 2, 16syl2anc 642 . . 3  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  ( H  o.  F )  Fn  A
)
18 simp3 957 . . . 4  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  K  Fn  B )
19 fnfco 5423 . . . 4  |-  ( ( K  Fn  B  /\  F : A --> B )  ->  ( K  o.  F )  Fn  A
)
2018, 2, 19syl2anc 642 . . 3  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  ( K  o.  F )  Fn  A
)
21 eqfnfv 5638 . . 3  |-  ( ( ( H  o.  F
)  Fn  A  /\  ( K  o.  F
)  Fn  A )  ->  ( ( H  o.  F )  =  ( K  o.  F
)  <->  A. y  e.  A  ( ( H  o.  F ) `  y
)  =  ( ( K  o.  F ) `
 y ) ) )
2217, 20, 21syl2anc 642 . 2  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  ( ( H  o.  F )  =  ( K  o.  F )  <->  A. y  e.  A  ( ( H  o.  F ) `  y )  =  ( ( K  o.  F
) `  y )
) )
23 eqfnfv 5638 . . 3  |-  ( ( H  Fn  B  /\  K  Fn  B )  ->  ( H  =  K  <->  A. x  e.  B  ( H `  x )  =  ( K `  x ) ) )
2415, 18, 23syl2anc 642 . 2  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  ( H  =  K  <->  A. x  e.  B  ( H `  x )  =  ( K `  x ) ) )
2514, 22, 243bitr4d 276 1  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  ( ( H  o.  F )  =  ( K  o.  F )  <->  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   -onto->wfo 5269   ` cfv 5271
This theorem is referenced by:  mapen  7041  mapfien  7415  hashfacen  11408  setcepi  13936  qtopeu  17423  qtophmeo  17524  derangenlem  23717  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-fo 5277  df-fv 5279
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