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Theorem cbvexfo 6015
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 6014 . . 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 2710 . 2  |-  ( E. x  e.  A  ph  <->  -. 
A. x  e.  A  -.  ph )
6 dfrex2 2710 . 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 1652   A.wral 2697   E.wrex 2698   -onto->wfo 5444   ` cfv 5446
This theorem is referenced by:  f1oweALT  6066  deg1ldg  20007
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fo 5452  df-fv 5454
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