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Theorem cbvfo 5546
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 5219 . . 3  |-  ( F : A -onto-> B  ->  F  Fn  A )
2 cbvfo.1 . . . . . 6  |-  ( ( F `  x )  =  y  ->  ( ph 
<->  ps ) )
32bicomd 139 . . . . 5  |-  ( ( F `  x )  =  y  ->  ( ps 
<-> 
ph ) )
43eqcoms 2091 . . . 4  |-  ( y  =  ( F `  x )  ->  ( ps 
<-> 
ph ) )
54ralrn 5421 . . 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 5220 . . 3  |-  ( F : A -onto-> B  ->  ran  F  =  B )
87raleqdv 2568 . 2  |-  ( F : A -onto-> B  -> 
( A. y  e. 
ran  F ps  <->  A. y  e.  B  ps )
)
96, 8bitr3d 188 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 103    = wceq 1289   A.wral 2359   ran crn 4429    Fn wfn 4997   -onto->wfo 5000   ` cfv 5002
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fo 5008  df-fv 5010
This theorem is referenced by:  cocan2  5549  supisolem  6682
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