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Theorem cbvfo 5761
Description: Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
Hypothesis
Ref Expression
cbvfo.1  |-  ( ( F `  x )  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvfo  |-  ( F : A -onto-> B  -> 
( A. x  e.  A  ph  <->  A. y  e.  B  ps )
)
Distinct variable groups:    x, y, A   
y, B    x, F, y    ph, y    ps, x
Allowed substitution hints:    ph( x)    ps( y)    B( x)

Proof of Theorem cbvfo
StepHypRef Expression
1 fofn 5420 . . 3  |-  ( F : A -onto-> B  ->  F  Fn  A )
2 cbvfo.1 . . . . . 6  |-  ( ( F `  x )  =  y  ->  ( ph 
<->  ps ) )
32bicomd 140 . . . . 5  |-  ( ( F `  x )  =  y  ->  ( ps 
<-> 
ph ) )
43eqcoms 2173 . . . 4  |-  ( y  =  ( F `  x )  ->  ( ps 
<-> 
ph ) )
54ralrn 5631 . . 3  |-  ( F  Fn  A  ->  ( A. y  e.  ran  F ps  <->  A. x  e.  A  ph ) )
61, 5syl 14 . 2  |-  ( F : A -onto-> B  -> 
( A. y  e. 
ran  F ps  <->  A. x  e.  A  ph ) )
7 forn 5421 . . 3  |-  ( F : A -onto-> B  ->  ran  F  =  B )
87raleqdv 2671 . 2  |-  ( F : A -onto-> B  -> 
( A. y  e. 
ran  F ps  <->  A. y  e.  B  ps )
)
96, 8bitr3d 189 1  |-  ( F : A -onto-> B  -> 
( A. x  e.  A  ph  <->  A. y  e.  B  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1348   A.wral 2448   ran crn 4610    Fn wfn 5191   -onto->wfo 5194   ` cfv 5196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-fo 5202  df-fv 5204
This theorem is referenced by:  cocan2  5764  supisolem  6981
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