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Theorem funimaexg 5472
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 5141 . . . . 5  |-  ( w  =  B  ->  ( A " w )  =  ( A " B
) )
21eleq1d 2455 . . . 4  |-  ( w  =  B  ->  (
( A " w
)  e.  _V  <->  ( A " B )  e.  _V ) )
32imbi2d 308 . . 3  |-  ( w  =  B  ->  (
( Fun  A  ->  ( A " w )  e.  _V )  <->  ( Fun  A  ->  ( A " B )  e.  _V ) ) )
4 dffun5 5409 . . . . 5  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x E. z A. y ( <. x ,  y >.  e.  A  ->  y  =  z ) ) )
54simprbi 451 . . . 4  |-  ( Fun 
A  ->  A. x E. z A. y (
<. x ,  y >.  e.  A  ->  y  =  z ) )
6 nfv 1626 . . . . . 6  |-  F/ z
<. x ,  y >.  e.  A
76axrep4 4267 . . . . 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 2905 . . . . . 6  |-  ( ( A " w )  e.  _V  <->  E. z 
z  =  ( A
" w ) )
9 dfima3 5148 . . . . . . . . 9  |-  ( A
" w )  =  { y  |  E. x ( x  e.  w  /\  <. x ,  y >.  e.  A
) }
109eqeq2i 2399 . . . . . . . 8  |-  ( z  =  ( A "
w )  <->  z  =  { y  |  E. x ( x  e.  w  /\  <. x ,  y >.  e.  A
) } )
11 abeq2 2494 . . . . . . . 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 241 . . . . . . 7  |-  ( z  =  ( A "
w )  <->  A. y
( y  e.  z  <->  E. x ( x  e.  w  /\  <. x ,  y >.  e.  A
) ) )
1312exbii 1589 . . . . . 6  |-  ( E. z  z  =  ( A " w )  <->  E. z A. y ( y  e.  z  <->  E. x
( x  e.  w  /\  <. x ,  y
>.  e.  A ) ) )
148, 13bitri 241 . . . . 5  |-  ( ( A " w )  e.  _V  <->  E. z A. y ( y  e.  z  <->  E. x ( x  e.  w  /\  <. x ,  y >.  e.  A
) ) )
157, 14sylibr 204 . . . 4  |-  ( A. x E. z A. y
( <. x ,  y
>.  e.  A  ->  y  =  z )  -> 
( A " w
)  e.  _V )
165, 15syl 16 . . 3  |-  ( Fun 
A  ->  ( A " w )  e.  _V )
173, 16vtoclg 2956 . 2  |-  ( B  e.  C  ->  ( Fun  A  ->  ( A " B )  e.  _V ) )
1817impcom 420 1  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A " B )  e. 
_V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546   E.wex 1547    = wceq 1649    e. wcel 1717   {cab 2375   _Vcvv 2901   <.cop 3762   "cima 4823   Rel wrel 4825   Fun wfun 5390
This theorem is referenced by:  funimaex  5473  resfunexg  5898  resfunexgALT  5899  fnexALT  5903  wdomimag  7490  carduniima  7912  dfac12lem2  7959  ttukeylem3  8326  nnexALT  9936  seqex  11254  fbasrn  17839  elfm3  17905  nobndlem1  25372
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 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pr 4346
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-br 4156  df-opab 4210  df-id 4441  df-xp 4826  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-fun 5398
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