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Theorem cbvfo 6167
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 5779 . . 3  |-  ( F : A -onto-> B  ->  F  Fn  A )
2 cbvfo.1 . . . . . 6  |-  ( ( F `  x )  =  y  ->  ( ph 
<->  ps ) )
32bicomd 201 . . . . 5  |-  ( ( F `  x )  =  y  ->  ( ps 
<-> 
ph ) )
43eqcoms 2466 . . . 4  |-  ( y  =  ( F `  x )  ->  ( ps 
<-> 
ph ) )
54ralrn 6010 . . 3  |-  ( F  Fn  A  ->  ( A. y  e.  ran  F ps  <->  A. x  e.  A  ph ) )
61, 5syl 16 . 2  |-  ( F : A -onto-> B  -> 
( A. y  e. 
ran  F ps  <->  A. x  e.  A  ph ) )
7 forn 5780 . . 3  |-  ( F : A -onto-> B  ->  ran  F  =  B )
87raleqdv 3057 . 2  |-  ( F : A -onto-> B  -> 
( A. y  e. 
ran  F ps  <->  A. y  e.  B  ps )
)
96, 8bitr3d 255 1  |-  ( F : A -onto-> B  -> 
( A. x  e.  A  ph  <->  A. y  e.  B  ps )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1398   A.wral 2804   ran crn 4989    Fn wfn 5565   -onto->wfo 5568   ` cfv 5570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fo 5576  df-fv 5578
This theorem is referenced by:  cbvexfo  6168  cocan2  6170  f1oweALT  6757  supisolem  7923  qtopeu  20383  deg1leb  22661  dchrelbas4  23716  cnpcon  28939  cocanfo  30448
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