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Theorem cbvfo 6178
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 5795 . . 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 2479 . . . 4  |-  ( y  =  ( F `  x )  ->  ( ps 
<-> 
ph ) )
54ralrn 6022 . . 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 5796 . . 3  |-  ( F : A -onto-> B  ->  ran  F  =  B )
87raleqdv 3064 . 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 1379   A.wral 2814   ran crn 5000    Fn wfn 5581   -onto->wfo 5584   ` cfv 5586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fo 5592  df-fv 5594
This theorem is referenced by:  cbvexfo  6179  cocan2  6181  f1oweALT  6765  supisolem  7927  qtopeu  19949  deg1leb  22227  dchrelbas4  23243  cnpcon  28312  cocanfo  29809
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