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Theorem cocan2 6194
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 5801 . . . . . . 7  |-  ( F : A -onto-> B  ->  F : A --> B )
213ad2ant1 1017 . . . . . 6  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  F : A
--> B )
3 fvco3 5951 . . . . . 6  |-  ( ( F : A --> B  /\  y  e.  A )  ->  ( ( H  o.  F ) `  y
)  =  ( H `
 ( F `  y ) ) )
42, 3sylan 471 . . . . 5  |-  ( ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  /\  y  e.  A )  ->  (
( H  o.  F
) `  y )  =  ( H `  ( F `  y ) ) )
5 fvco3 5951 . . . . . 6  |-  ( ( F : A --> B  /\  y  e.  A )  ->  ( ( K  o.  F ) `  y
)  =  ( K `
 ( F `  y ) ) )
62, 5sylan 471 . . . . 5  |-  ( ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  /\  y  e.  A )  ->  (
( K  o.  F
) `  y )  =  ( K `  ( F `  y ) ) )
74, 6eqeq12d 2489 . . . 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 2903 . . 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 5872 . . . . . 6  |-  ( ( F `  y )  =  x  ->  ( H `  ( F `  y ) )  =  ( H `  x
) )
10 fveq2 5872 . . . . . 6  |-  ( ( F `  y )  =  x  ->  ( K `  ( F `  y ) )  =  ( K `  x
) )
119, 10eqeq12d 2489 . . . . 5  |-  ( ( F `  y )  =  x  ->  (
( H `  ( F `  y )
)  =  ( K `
 ( F `  y ) )  <->  ( H `  x )  =  ( K `  x ) ) )
1211cbvfo 6191 . . . 4  |-  ( F : A -onto-> B  -> 
( A. y  e.  A  ( H `  ( F `  y ) )  =  ( K `
 ( F `  y ) )  <->  A. x  e.  B  ( H `  x )  =  ( K `  x ) ) )
13123ad2ant1 1017 . . 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 253 . 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 997 . . . 4  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  H  Fn  B )
16 fnfco 5756 . . . 4  |-  ( ( H  Fn  B  /\  F : A --> B )  ->  ( H  o.  F )  Fn  A
)
1715, 2, 16syl2anc 661 . . 3  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  ( H  o.  F )  Fn  A
)
18 simp3 998 . . . 4  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  K  Fn  B )
19 fnfco 5756 . . . 4  |-  ( ( K  Fn  B  /\  F : A --> B )  ->  ( K  o.  F )  Fn  A
)
2018, 2, 19syl2anc 661 . . 3  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  ( K  o.  F )  Fn  A
)
21 eqfnfv 5982 . . 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 661 . 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 5982 . . 3  |-  ( ( H  Fn  B  /\  K  Fn  B )  ->  ( H  =  K  <->  A. x  e.  B  ( H `  x )  =  ( K `  x ) ) )
2415, 18, 23syl2anc 661 . 2  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  ( H  =  K  <->  A. x  e.  B  ( H `  x )  =  ( K `  x ) ) )
2514, 22, 243bitr4d 285 1  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  ( ( H  o.  F )  =  ( K  o.  F )  <->  H  =  K ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817    o. ccom 5009    Fn wfn 5589   -->wf 5590   -onto->wfo 5592   ` cfv 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fo 5600  df-fv 5602
This theorem is referenced by:  mapen  7693  mapfien  7879  mapfienOLD  8150  hashfacen  12484  setcepi  15290  qtopeu  20085  qtophmeo  20186  derangenlem  28440
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