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Theorem cbvfo 5799
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 5453 . . 3  |-  ( F : A -onto-> B  ->  F  Fn  A )
2 cbvfo.1 . . . . . 6  |-  ( ( F `  x )  =  y  ->  ( ph 
<->  ps ) )
32bicomd 192 . . . . 5  |-  ( ( F `  x )  =  y  ->  ( ps 
<-> 
ph ) )
43eqcoms 2286 . . . 4  |-  ( y  =  ( F `  x )  ->  ( ps 
<-> 
ph ) )
54ralrn 5668 . . 3  |-  ( F  Fn  A  ->  ( A. y  e.  ran  F ps  <->  A. x  e.  A  ph ) )
61, 5syl 15 . 2  |-  ( F : A -onto-> B  -> 
( A. y  e. 
ran  F ps  <->  A. x  e.  A  ph ) )
7 forn 5454 . . 3  |-  ( F : A -onto-> B  ->  ran  F  =  B )
87raleqdv 2742 . 2  |-  ( F : A -onto-> B  -> 
( A. y  e. 
ran  F ps  <->  A. y  e.  B  ps )
)
96, 8bitr3d 246 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 176    = wceq 1623   A.wral 2543   ran crn 4690    Fn wfn 5250   -onto->wfo 5253   ` cfv 5255
This theorem is referenced by:  cbvexfo  5800  cocan2  5802  f1oweALT  5851  supisolem  7221  qtopeu  17407  deg1leb  19481  dchrelbas4  20482  cnpcon  23761  cocanfo  26374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263
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