Theorem funimaexg 5272

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

Step	Hyp	Ref	Expression
1		simpl 108	. . 3 |- ( ( Fun A /\ B e. C ) -> Fun A )
2		funrel 5205	. . 3 |- ( Fun A -> Rel A )
3		resres 4896	. . . . . . 7 |- ( ( A |` dom A ) |` B ) = ( A |` ( dom A i^i B ) )
4		incom 3314	. . . . . . . 8 |- ( B i^i dom A ) = ( dom A i^i B )
5	4	reseq2i 4881	. . . . . . 7 |- ( A |` ( B i^i dom A ) ) = ( A |` ( dom A i^i B ) )
6	3, 5	eqtr4i 2189	. . . . . 6 |- ( ( A |` dom A ) |` B ) = ( A |` ( B i^i dom A ) )
7		resdm 4923	. . . . . . 7 |- ( Rel A -> ( A |` dom A ) = A )
8	7	reseq1d 4883	. . . . . 6 |- ( Rel A -> ( ( A |` dom A ) |` B ) = ( A |` B ) )
9	6, 8	eqtr3id 2213	. . . . 5 |- ( Rel A -> ( A |` ( B i^i dom A ) ) = ( A |` B ) )
10	9	rneqd 4833	. . . 4 |- ( Rel A -> ran ( A |` ( B i^i dom A ) ) = ran ( A |` B ) )
11		df-ima 4617	. . . 4 |- ( A " ( B i^i dom A ) ) = ran ( A |` ( B i^i dom A ) )
12		df-ima 4617	. . . 4 |- ( A " B ) = ran ( A |` B )
13	10, 11, 12	3eqtr4g 2224	. . 3 |- ( Rel A -> ( A " ( B i^i dom A ) ) = ( A " B ) )
14	1, 2, 13	3syl 17	. 2 |- ( ( Fun A /\ B e. C ) -> ( A " ( B i^i dom A ) ) = ( A " B ) )
15		inex1g 4118	. . 3 |- ( B e. C -> ( B i^i dom A ) e. _V )
16		inss2 3343	. . . 4 |- ( B i^i dom A ) C_ dom A
17		funimaexglem 5271	. . . 4 |- ( ( Fun A /\ ( B i^i dom A ) e. _V /\ ( B i^i dom A ) C_ dom A ) -> ( A " ( B i^i dom A ) ) e. _V )
18	16, 17	mp3an3 1316	. . 3 |- ( ( Fun A /\ ( B i^i dom A ) e. _V ) -> ( A " ( B i^i dom A ) ) e. _V )
19	15, 18	sylan2 284	. 2 |- ( ( Fun A /\ B e. C ) -> ( A " ( B i^i dom A ) ) e. _V )
20	14, 19	eqeltrrd 2244	1 |- ( ( Fun A /\ B e. C ) -> ( A " B ) e. _V )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   _Vcvv 2726   i^i cin 3115   C_ wss 3116   dom cdm 4604   ran crn 4605   |` cres 4606   "cima 4607   Rel wrel 4609   Fun wfun 5182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-fun 5190

This theorem is referenced by:  funimaex  5273  resfunexg  5706  resfunexgALT  6076  fnexALT  6079  suplocexprlem2b  7655  suplocexprlemlub  7665