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Theorem cocan2 5453
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 5131 . . . . . . 7  |-  ( F : A -onto-> B  ->  F : A --> B )
213ad2ant1 960 . . . . . 6  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  F : A
--> B )
3 fvco3 5270 . . . . . 6  |-  ( ( F : A --> B  /\  y  e.  A )  ->  ( ( H  o.  F ) `  y
)  =  ( H `
 ( F `  y ) ) )
42, 3sylan 277 . . . . 5  |-  ( ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  /\  y  e.  A )  ->  (
( H  o.  F
) `  y )  =  ( H `  ( F `  y ) ) )
5 fvco3 5270 . . . . . 6  |-  ( ( F : A --> B  /\  y  e.  A )  ->  ( ( K  o.  F ) `  y
)  =  ( K `
 ( F `  y ) ) )
62, 5sylan 277 . . . . 5  |-  ( ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  /\  y  e.  A )  ->  (
( K  o.  F
) `  y )  =  ( K `  ( F `  y ) ) )
74, 6eqeq12d 2096 . . . 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 2365 . . 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 5203 . . . . . 6  |-  ( ( F `  y )  =  x  ->  ( H `  ( F `  y ) )  =  ( H `  x
) )
10 fveq2 5203 . . . . . 6  |-  ( ( F `  y )  =  x  ->  ( K `  ( F `  y ) )  =  ( K `  x
) )
119, 10eqeq12d 2096 . . . . 5  |-  ( ( F `  y )  =  x  ->  (
( H `  ( F `  y )
)  =  ( K `
 ( F `  y ) )  <->  ( H `  x )  =  ( K `  x ) ) )
1211cbvfo 5450 . . . 4  |-  ( F : A -onto-> B  -> 
( A. y  e.  A  ( H `  ( F `  y ) )  =  ( K `
 ( F `  y ) )  <->  A. x  e.  B  ( H `  x )  =  ( K `  x ) ) )
13123ad2ant1 960 . . 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 186 . 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 940 . . . 4  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  H  Fn  B )
16 fnfco 5090 . . . 4  |-  ( ( H  Fn  B  /\  F : A --> B )  ->  ( H  o.  F )  Fn  A
)
1715, 2, 16syl2anc 403 . . 3  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  ( H  o.  F )  Fn  A
)
18 simp3 941 . . . 4  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  K  Fn  B )
19 fnfco 5090 . . . 4  |-  ( ( K  Fn  B  /\  F : A --> B )  ->  ( K  o.  F )  Fn  A
)
2018, 2, 19syl2anc 403 . . 3  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  ( K  o.  F )  Fn  A
)
21 eqfnfv 5291 . . 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 403 . 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 5291 . . 3  |-  ( ( H  Fn  B  /\  K  Fn  B )  ->  ( H  =  K  <->  A. x  e.  B  ( H `  x )  =  ( K `  x ) ) )
2415, 18, 23syl2anc 403 . 2  |-  ( ( F : A -onto-> B  /\  H  Fn  B  /\  K  Fn  B
)  ->  ( H  =  K  <->  A. x  e.  B  ( H `  x )  =  ( K `  x ) ) )
2514, 22, 243bitr4d 218 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    /\ wa 102    <-> wb 103    /\ w3a 920    = wceq 1285    e. wcel 1434   A.wral 2349    o. ccom 4369    Fn wfn 4921   -->wf 4922   -onto->wfo 4924   ` cfv 4926
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-csb 2910  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-br 3788  df-opab 3842  df-mpt 3843  df-id 4050  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-fo 4932  df-fv 4934
This theorem is referenced by: (None)
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