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Theorem cbvexfo 6095
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 294 . . . 4  |-  ( ( F `  x )  =  y  ->  ( -.  ph  <->  -.  ps )
)
32cbvfo 6094 . . 3  |-  ( F : A -onto-> B  -> 
( A. x  e.  A  -.  ph  <->  A. y  e.  B  -.  ps )
)
43notbid 294 . 2  |-  ( F : A -onto-> B  -> 
( -.  A. x  e.  A  -.  ph  <->  -.  A. y  e.  B  -.  ps )
)
5 dfrex2 2870 . 2  |-  ( E. x  e.  A  ph  <->  -. 
A. x  e.  A  -.  ph )
6 dfrex2 2870 . 2  |-  ( E. y  e.  B  ps  <->  -. 
A. y  e.  B  -.  ps )
74, 5, 63bitr4g 288 1  |-  ( F : A -onto-> B  -> 
( E. x  e.  A  ph  <->  E. y  e.  B  ps )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    = wceq 1370   A.wral 2795   E.wrex 2796   -onto->wfo 5516   ` cfv 5518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-fo 5524  df-fv 5526
This theorem is referenced by:  f1oweALT  6663  deg1ldg  21681
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