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Theorem funimaexg 5646
Description: Axiom of Replacement using abbreviations. Axiom 39(vi) of [Quine] p. 284. Compare Exercise 9 of [TakeutiZaring] p. 29. (Contributed by NM, 10-Sep-2006.)
Assertion
Ref Expression
funimaexg  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A " B )  e. 
_V )

Proof of Theorem funimaexg
Dummy variables  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imaeq2 5153 . . . . 5  |-  ( w  =  B  ->  ( A " w )  =  ( A " B
) )
21eleq1d 2471 . . . 4  |-  ( w  =  B  ->  (
( A " w
)  e.  _V  <->  ( A " B )  e.  _V ) )
32imbi2d 314 . . 3  |-  ( w  =  B  ->  (
( Fun  A  ->  ( A " w )  e.  _V )  <->  ( Fun  A  ->  ( A " B )  e.  _V ) ) )
4 dffun5 5582 . . . . 5  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x E. z A. y ( <. x ,  y >.  e.  A  ->  y  =  z ) ) )
54simprbi 462 . . . 4  |-  ( Fun 
A  ->  A. x E. z A. y (
<. x ,  y >.  e.  A  ->  y  =  z ) )
6 nfv 1728 . . . . . 6  |-  F/ z
<. x ,  y >.  e.  A
76axrep4 4511 . . . . 5  |-  ( A. x E. z A. y
( <. x ,  y
>.  e.  A  ->  y  =  z )  ->  E. z A. y ( y  e.  z  <->  E. x
( x  e.  w  /\  <. x ,  y
>.  e.  A ) ) )
8 isset 3063 . . . . . 6  |-  ( ( A " w )  e.  _V  <->  E. z 
z  =  ( A
" w ) )
9 dfima3 5160 . . . . . . . . 9  |-  ( A
" w )  =  { y  |  E. x ( x  e.  w  /\  <. x ,  y >.  e.  A
) }
109eqeq2i 2420 . . . . . . . 8  |-  ( z  =  ( A "
w )  <->  z  =  { y  |  E. x ( x  e.  w  /\  <. x ,  y >.  e.  A
) } )
11 abeq2 2526 . . . . . . . 8  |-  ( z  =  { y  |  E. x ( x  e.  w  /\  <. x ,  y >.  e.  A
) }  <->  A. y
( y  e.  z  <->  E. x ( x  e.  w  /\  <. x ,  y >.  e.  A
) ) )
1210, 11bitri 249 . . . . . . 7  |-  ( z  =  ( A "
w )  <->  A. y
( y  e.  z  <->  E. x ( x  e.  w  /\  <. x ,  y >.  e.  A
) ) )
1312exbii 1688 . . . . . 6  |-  ( E. z  z  =  ( A " w )  <->  E. z A. y ( y  e.  z  <->  E. x
( x  e.  w  /\  <. x ,  y
>.  e.  A ) ) )
148, 13bitri 249 . . . . 5  |-  ( ( A " w )  e.  _V  <->  E. z A. y ( y  e.  z  <->  E. x ( x  e.  w  /\  <. x ,  y >.  e.  A
) ) )
157, 14sylibr 212 . . . 4  |-  ( A. x E. z A. y
( <. x ,  y
>.  e.  A  ->  y  =  z )  -> 
( A " w
)  e.  _V )
165, 15syl 17 . . 3  |-  ( Fun 
A  ->  ( A " w )  e.  _V )
173, 16vtoclg 3117 . 2  |-  ( B  e.  C  ->  ( Fun  A  ->  ( A " B )  e.  _V ) )
1817impcom 428 1  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A " B )  e. 
_V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1403    = wceq 1405   E.wex 1633    e. wcel 1842   {cab 2387   _Vcvv 3059   <.cop 3978   "cima 4826   Rel wrel 4828   Fun wfun 5563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-br 4396  df-opab 4454  df-id 4738  df-xp 4829  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-fun 5571
This theorem is referenced by:  funimaex  5647  resfunexg  6118  resfunexgALT  6747  fnexALT  6750  wdomimag  8047  carduniima  8509  dfac12lem2  8556  ttukeylem3  8923  nnexALT  10578  seqex  12153  fbasrn  20677  elfm3  20743  nobndlem1  30152
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