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Theorem cbvexfo 5680
Description: Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.)
Hypothesis
Ref Expression
cbvfo.1  |-  ( ( F `  x )  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvexfo  |-  ( F : A -onto-> B  -> 
( E. x  e.  A  ph  <->  E. 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 cbvexfo
StepHypRef Expression
1 fofn 5342 . . 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 2140 . . . 4  |-  ( y  =  ( F `  x )  ->  ( ps 
<-> 
ph ) )
54rexrn 5550 . . 3  |-  ( F  Fn  A  ->  ( E. y  e.  ran  F ps  <->  E. x  e.  A  ph ) )
61, 5syl 14 . 2  |-  ( F : A -onto-> B  -> 
( E. y  e. 
ran  F ps  <->  E. x  e.  A  ph ) )
7 forn 5343 . . 3  |-  ( F : A -onto-> B  ->  ran  F  =  B )
87rexeqdv 2631 . 2  |-  ( F : A -onto-> B  -> 
( E. y  e. 
ran  F ps  <->  E. y  e.  B  ps )
)
96, 8bitr3d 189 1  |-  ( F : A -onto-> B  -> 
( E. x  e.  A  ph  <->  E. y  e.  B  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1331   E.wrex 2415   ran crn 4535    Fn wfn 5113   -onto->wfo 5116   ` cfv 5118
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fo 5124  df-fv 5126
This theorem is referenced by: (None)
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