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Theorem funimaexg 5522
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 5191 . . . . 5  |-  ( w  =  B  ->  ( A " w )  =  ( A " B
) )
21eleq1d 2501 . . . 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 5459 . . . . 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 1629 . . . . . 6  |-  F/ z
<. x ,  y >.  e.  A
76axrep4 4316 . . . . 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 2952 . . . . . 6  |-  ( ( A " w )  e.  _V  <->  E. z 
z  =  ( A
" w ) )
9 dfima3 5198 . . . . . . . . 9  |-  ( A
" w )  =  { y  |  E. x ( x  e.  w  /\  <. x ,  y >.  e.  A
) }
109eqeq2i 2445 . . . . . . . 8  |-  ( z  =  ( A "
w )  <->  z  =  { y  |  E. x ( x  e.  w  /\  <. x ,  y >.  e.  A
) } )
11 abeq2 2540 . . . . . . . 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 1592 . . . . . 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 3003 . 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 1549   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2421   _Vcvv 2948   <.cop 3809   "cima 4873   Rel wrel 4875   Fun wfun 5440
This theorem is referenced by:  funimaex  5523  resfunexg  5949  resfunexgALT  5950  fnexALT  5954  wdomimag  7547  carduniima  7969  dfac12lem2  8016  ttukeylem3  8383  nnexALT  9994  seqex  11317  fbasrn  17908  elfm3  17974  nobndlem1  25639
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-rep 4312  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-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-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-fun 5448
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