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Theorem cbvexfo 5816
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 285 . . . 4  |-  ( ( F `  x )  =  y  ->  ( -.  ph  <->  -.  ps )
)
32cbvfo 5815 . . 3  |-  ( F : A -onto-> B  -> 
( A. x  e.  A  -.  ph  <->  A. y  e.  B  -.  ps )
)
43notbid 285 . 2  |-  ( F : A -onto-> B  -> 
( -.  A. x  e.  A  -.  ph  <->  -.  A. y  e.  B  -.  ps )
)
5 dfrex2 2569 . 2  |-  ( E. x  e.  A  ph  <->  -. 
A. x  e.  A  -.  ph )
6 dfrex2 2569 . 2  |-  ( E. y  e.  B  ps  <->  -. 
A. y  e.  B  -.  ps )
74, 5, 63bitr4g 279 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 176    = wceq 1632   A.wral 2556   E.wrex 2557   -onto->wfo 5269   ` cfv 5271
This theorem is referenced by:  f1oweALT  5867  deg1ldg  19494
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279
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