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Theorem cocan2 6027
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 5655 . . . . . . 7  |-  ( F : A -onto-> B  ->  F : A --> B )
213ad2ant1 979 . . . . . 6  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  F : A
--> B )
3 fvco3 5802 . . . . . 6  |-  ( ( F : A --> B  /\  y  e.  A )  ->  ( ( H  o.  F ) `  y
)  =  ( H `
 ( F `  y ) ) )
42, 3sylan 459 . . . . 5  |-  ( ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  /\  y  e.  A )  ->  (
( H  o.  F
) `  y )  =  ( H `  ( F `  y ) ) )
5 fvco3 5802 . . . . . 6  |-  ( ( F : A --> B  /\  y  e.  A )  ->  ( ( K  o.  F ) `  y
)  =  ( K `
 ( F `  y ) ) )
62, 5sylan 459 . . . . 5  |-  ( ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  /\  y  e.  A )  ->  (
( K  o.  F
) `  y )  =  ( K `  ( F `  y ) ) )
74, 6eqeq12d 2452 . . . 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 2723 . . 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 5730 . . . . . 6  |-  ( ( F `  y )  =  x  ->  ( H `  ( F `  y ) )  =  ( H `  x
) )
10 fveq2 5730 . . . . . 6  |-  ( ( F `  y )  =  x  ->  ( K `  ( F `  y ) )  =  ( K `  x
) )
119, 10eqeq12d 2452 . . . . 5  |-  ( ( F `  y )  =  x  ->  (
( H `  ( F `  y )
)  =  ( K `
 ( F `  y ) )  <->  ( H `  x )  =  ( K `  x ) ) )
1211cbvfo 6024 . . . 4  |-  ( F : A -onto-> B  -> 
( A. y  e.  A  ( H `  ( F `  y ) )  =  ( K `
 ( F `  y ) )  <->  A. x  e.  B  ( H `  x )  =  ( K `  x ) ) )
13123ad2ant1 979 . . 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 246 . 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 959 . . . 4  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  H  Fn  B )
16 fnfco 5611 . . . 4  |-  ( ( H  Fn  B  /\  F : A --> B )  ->  ( H  o.  F )  Fn  A
)
1715, 2, 16syl2anc 644 . . 3  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  ( H  o.  F )  Fn  A
)
18 simp3 960 . . . 4  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  K  Fn  B )
19 fnfco 5611 . . . 4  |-  ( ( K  Fn  B  /\  F : A --> B )  ->  ( K  o.  F )  Fn  A
)
2018, 2, 19syl2anc 644 . . 3  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  ( K  o.  F )  Fn  A
)
21 eqfnfv 5829 . . 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 644 . 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 5829 . . 3  |-  ( ( H  Fn  B  /\  K  Fn  B )  ->  ( H  =  K  <->  A. x  e.  B  ( H `  x )  =  ( K `  x ) ) )
2415, 18, 23syl2anc 644 . 2  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  ( H  =  K  <->  A. x  e.  B  ( H `  x )  =  ( K `  x ) ) )
2514, 22, 243bitr4d 278 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707    o. ccom 4884    Fn wfn 5451   -->wf 5452   -onto->wfo 5454   ` cfv 5456
This theorem is referenced by:  mapen  7273  mapfien  7655  hashfacen  11705  setcepi  14245  qtopeu  17750  qtophmeo  17851  derangenlem  24859
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fo 5462  df-fv 5464
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