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Theorem cbvexfo 5962
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 cbvfo.1 . . . . 5  |-  ( ( F `  x )  =  y  ->  ( ph 
<->  ps ) )
21notbid 286 . . . 4  |-  ( ( F `  x )  =  y  ->  ( -.  ph  <->  -.  ps )
)
32cbvfo 5961 . . 3  |-  ( F : A -onto-> B  -> 
( A. x  e.  A  -.  ph  <->  A. y  e.  B  -.  ps )
)
43notbid 286 . 2  |-  ( F : A -onto-> B  -> 
( -.  A. x  e.  A  -.  ph  <->  -.  A. y  e.  B  -.  ps )
)
5 dfrex2 2662 . 2  |-  ( E. x  e.  A  ph  <->  -. 
A. x  e.  A  -.  ph )
6 dfrex2 2662 . 2  |-  ( E. y  e.  B  ps  <->  -. 
A. y  e.  B  -.  ps )
74, 5, 63bitr4g 280 1  |-  ( F : A -onto-> B  -> 
( E. x  e.  A  ph  <->  E. y  e.  B  ps )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    = wceq 1649   A.wral 2649   E.wrex 2650   -onto->wfo 5392   ` cfv 5394
This theorem is referenced by:  f1oweALT  6013  deg1ldg  19882
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-fo 5400  df-fv 5402
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