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Theorem funimaexg 5345
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 5024 . . . . 5  |-  ( w  =  B  ->  ( A " w )  =  ( A " B
) )
21eleq1d 2362 . . . 4  |-  ( w  =  B  ->  (
( A " w
)  e.  _V  <->  ( A " B )  e.  _V ) )
32imbi2d 307 . . 3  |-  ( w  =  B  ->  (
( Fun  A  ->  ( A " w )  e.  _V )  <->  ( Fun  A  ->  ( A " B )  e.  _V ) ) )
4 dffun5 5284 . . . . 5  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x E. z A. y ( <. x ,  y >.  e.  A  ->  y  =  z ) ) )
54simprbi 450 . . . 4  |-  ( Fun 
A  ->  A. x E. z A. y (
<. x ,  y >.  e.  A  ->  y  =  z ) )
6 nfv 1609 . . . . . 6  |-  F/ z
<. x ,  y >.  e.  A
76axrep4 4151 . . . . 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 2805 . . . . . 6  |-  ( ( A " w )  e.  _V  <->  E. z 
z  =  ( A
" w ) )
9 dfima3 5031 . . . . . . . . 9  |-  ( A
" w )  =  { y  |  E. x ( x  e.  w  /\  <. x ,  y >.  e.  A
) }
109eqeq2i 2306 . . . . . . . 8  |-  ( z  =  ( A "
w )  <->  z  =  { y  |  E. x ( x  e.  w  /\  <. x ,  y >.  e.  A
) } )
11 abeq2 2401 . . . . . . . 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 240 . . . . . . 7  |-  ( z  =  ( A "
w )  <->  A. y
( y  e.  z  <->  E. x ( x  e.  w  /\  <. x ,  y >.  e.  A
) ) )
1312exbii 1572 . . . . . 6  |-  ( E. z  z  =  ( A " w )  <->  E. z A. y ( y  e.  z  <->  E. x
( x  e.  w  /\  <. x ,  y
>.  e.  A ) ) )
148, 13bitri 240 . . . . 5  |-  ( ( A " w )  e.  _V  <->  E. z A. y ( y  e.  z  <->  E. x ( x  e.  w  /\  <. x ,  y >.  e.  A
) ) )
157, 14sylibr 203 . . . 4  |-  ( A. x E. z A. y
( <. x ,  y
>.  e.  A  ->  y  =  z )  -> 
( A " w
)  e.  _V )
165, 15syl 15 . . 3  |-  ( Fun 
A  ->  ( A " w )  e.  _V )
173, 16vtoclg 2856 . 2  |-  ( B  e.  C  ->  ( Fun  A  ->  ( A " B )  e.  _V ) )
1817impcom 419 1  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A " B )  e. 
_V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282   _Vcvv 2801   <.cop 3656   "cima 4708   Rel wrel 4710   Fun wfun 5265
This theorem is referenced by:  funimaex  5346  resfunexg  5753  resfunexgALT  5754  fnexALT  5758  wdomimag  7317  carduniima  7739  dfac12lem2  7786  ttukeylem3  8154  nnexALT  9764  seqex  11064  fbasrn  17595  elfm3  17661  nobndlem1  24417
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-rep 4147  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-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-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-fun 5273
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