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Theorem funimaexg 5658
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 5326 . . . . 5  |-  ( w  =  B  ->  ( A " w )  =  ( A " B
) )
21eleq1d 2531 . . . 4  |-  ( w  =  B  ->  (
( A " w
)  e.  _V  <->  ( A " B )  e.  _V ) )
32imbi2d 316 . . 3  |-  ( w  =  B  ->  (
( Fun  A  ->  ( A " w )  e.  _V )  <->  ( Fun  A  ->  ( A " B )  e.  _V ) ) )
4 dffun5 5594 . . . . 5  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x E. z A. y ( <. x ,  y >.  e.  A  ->  y  =  z ) ) )
54simprbi 464 . . . 4  |-  ( Fun 
A  ->  A. x E. z A. y (
<. x ,  y >.  e.  A  ->  y  =  z ) )
6 nfv 1678 . . . . . 6  |-  F/ z
<. x ,  y >.  e.  A
76axrep4 4557 . . . . 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 3112 . . . . . 6  |-  ( ( A " w )  e.  _V  <->  E. z 
z  =  ( A
" w ) )
9 dfima3 5333 . . . . . . . . 9  |-  ( A
" w )  =  { y  |  E. x ( x  e.  w  /\  <. x ,  y >.  e.  A
) }
109eqeq2i 2480 . . . . . . . 8  |-  ( z  =  ( A "
w )  <->  z  =  { y  |  E. x ( x  e.  w  /\  <. x ,  y >.  e.  A
) } )
11 abeq2 2586 . . . . . . . 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 1639 . . . . . 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 16 . . 3  |-  ( Fun 
A  ->  ( A " w )  e.  _V )
173, 16vtoclg 3166 . 2  |-  ( B  e.  C  ->  ( Fun  A  ->  ( A " B )  e.  _V ) )
1817impcom 430 1  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A " B )  e. 
_V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1372    = wceq 1374   E.wex 1591    e. wcel 1762   {cab 2447   _Vcvv 3108   <.cop 4028   "cima 4997   Rel wrel 4999   Fun wfun 5575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-br 4443  df-opab 4501  df-id 4790  df-xp 5000  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-fun 5583
This theorem is referenced by:  funimaex  5659  resfunexg  6119  resfunexgALT  6739  fnexALT  6742  wdomimag  8004  carduniima  8468  dfac12lem2  8515  ttukeylem3  8882  nnexALT  10529  seqex  12067  fbasrn  20115  elfm3  20181  nobndlem1  29017
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