Theorem funimaexg 5421

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 109	. . 3 |- ( ( Fun A /\ B e. C ) -> Fun A )
2		funrel 5350	. . 3 |- ( Fun A -> Rel A )
3		resres 5031	. . . . . . 7 |- ( ( A |` dom A ) |` B ) = ( A |` ( dom A i^i B ) )
4		incom 3401	. . . . . . . 8 |- ( B i^i dom A ) = ( dom A i^i B )
5	4	reseq2i 5016	. . . . . . 7 |- ( A |` ( B i^i dom A ) ) = ( A |` ( dom A i^i B ) )
6	3, 5	eqtr4i 2255	. . . . . 6 |- ( ( A |` dom A ) |` B ) = ( A |` ( B i^i dom A ) )
7		resdm 5058	. . . . . . 7 |- ( Rel A -> ( A |` dom A ) = A )
8	7	reseq1d 5018	. . . . . 6 |- ( Rel A -> ( ( A |` dom A ) |` B ) = ( A |` B ) )
9	6, 8	eqtr3id 2278	. . . . 5 |- ( Rel A -> ( A |` ( B i^i dom A ) ) = ( A |` B ) )
10	9	rneqd 4967	. . . 4 |- ( Rel A -> ran ( A |` ( B i^i dom A ) ) = ran ( A |` B ) )
11		df-ima 4744	. . . 4 |- ( A " ( B i^i dom A ) ) = ran ( A |` ( B i^i dom A ) )
12		df-ima 4744	. . . 4 |- ( A " B ) = ran ( A |` B )
13	10, 11, 12	3eqtr4g 2289	. . 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 4230	. . 3 |- ( B e. C -> ( B i^i dom A ) e. _V )
16		inss2 3430	. . . 4 |- ( B i^i dom A ) C_ dom A
17		funimaexglem 5420	. . . 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 1363	. . 3 |- ( ( Fun A /\ ( B i^i dom A ) e. _V ) -> ( A " ( B i^i dom A ) ) e. _V )
19	15, 18	sylan2 286	. 2 |- ( ( Fun A /\ B e. C ) -> ( A " ( B i^i dom A ) ) e. _V )
20	14, 19	eqeltrrd 2309	1 |- ( ( Fun A /\ B e. C ) -> ( A " B ) e. _V )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   _Vcvv 2803   i^i cin 3200   C_ wss 3201   dom cdm 4731   ran crn 4732   |` cres 4733   "cima 4734   Rel wrel 4736   Fun wfun 5327

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-fun 5335

This theorem is referenced by:  funimaex  5422  resfunexg  5883  resfunexgALT  6279  fnexALT  6282  suplocexprlem2b  7977  suplocexprlemlub  7987