Theorem funimaexg 5975

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.

Step	Hyp	Ref	Expression
1		imaeq2 5462	. . . . 5 |- ( w = B -> ( A " w ) = ( A " B ) )
2	1	eleq1d 2686	. . . 4 |- ( w = B -> ( ( A " w ) e. _V <-> ( A " B ) e. _V ) )
3	2	imbi2d 330	. . 3 |- ( w = B -> ( ( Fun A -> ( A " w ) e. _V ) <-> ( Fun A -> ( A " B ) e. _V ) ) )
4		dffun5 5901	. . . . 5 |- ( Fun A <-> ( Rel A /\ A. x E. z A. y ( <. x , y >. e. A -> y = z ) ) )
5	4	simprbi 480	. . . 4 |- ( Fun A -> A. x E. z A. y ( <. x , y >. e. A -> y = z ) )
6		nfv 1843	. . . . . 6 |- F/ z <. x , y >. e. A
7	6	axrep4 4775	. . . . 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 3207	. . . . . 6 |- ( ( A " w ) e. _V <-> E. z z = ( A " w ) )
9		dfima3 5469	. . . . . . . . 9 |- ( A " w ) = { y | E. x ( x e. w /\ <. x , y >. e. A ) }
10	9	eqeq2i 2634	. . . . . . . 8 |- ( z = ( A " w ) <-> z = { y | E. x ( x e. w /\ <. x , y >. e. A ) } )
11		abeq2 2732	. . . . . . . 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 ) ) )
12	10, 11	bitri 264	. . . . . . 7 |- ( z = ( A " w ) <-> A. y ( y e. z <-> E. x ( x e. w /\ <. x , y >. e. A ) ) )
13	12	exbii 1774	. . . . . 6 |- ( E. z z = ( A " w ) <-> E. z A. y ( y e. z <-> E. x ( x e. w /\ <. x , y >. e. A ) ) )
14	8, 13	bitri 264	. . . . 5 |- ( ( A " w ) e. _V <-> E. z A. y ( y e. z <-> E. x ( x e. w /\ <. x , y >. e. A ) ) )
15	7, 14	sylibr 224	. . . 4 |- ( A. x E. z A. y ( <. x , y >. e. A -> y = z ) -> ( A " w ) e. _V )
16	5, 15	syl 17	. . 3 |- ( Fun A -> ( A " w ) e. _V )
17	3, 16	vtoclg 3266	. 2 |- ( B e. C -> ( Fun A -> ( A " B ) e. _V ) )
18	17	impcom 446	1 |- ( ( Fun A /\ B e. C ) -> ( A " B ) e. _V )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   _Vcvv 3200   <.cop 4183   "cima 5117   Rel wrel 5119   Fun wfun 5882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-fun 5890

This theorem is referenced by:  funimaex  5976  resfunexg  6479  resfunexgALT  7129  fnexALT  7132  wdomimag  8492  carduniima  8919  dfac12lem2  8966  ttukeylem3  9333  nnexALT  11022  seqex  12803  fbasrn  21688  elfm3  21754  bdayimaon  31843  nosupno  31849  madeval  31935