Theorem funimaexg 5460

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 5389	. . 3 |- ( Fun A -> Rel A )
3		resres 5070	. . . . . . 7 |- ( ( A |` dom A ) |` B ) = ( A |` ( dom A i^i B ) )
4		incom 3421	. . . . . . . 8 |- ( B i^i dom A ) = ( dom A i^i B )
5	4	reseq2i 5055	. . . . . . 7 |- ( A |` ( B i^i dom A ) ) = ( A |` ( dom A i^i B ) )
6	3, 5	eqtr4i 2262	. . . . . 6 |- ( ( A |` dom A ) |` B ) = ( A |` ( B i^i dom A ) )
7		resdm 5097	. . . . . . 7 |- ( Rel A -> ( A |` dom A ) = A )
8	7	reseq1d 5057	. . . . . 6 |- ( Rel A -> ( ( A |` dom A ) |` B ) = ( A |` B ) )
9	6, 8	eqtr3id 2285	. . . . 5 |- ( Rel A -> ( A |` ( B i^i dom A ) ) = ( A |` B ) )
10	9	rneqd 5006	. . . 4 |- ( Rel A -> ran ( A |` ( B i^i dom A ) ) = ran ( A |` B ) )
11		df-ima 4782	. . . 4 |- ( A " ( B i^i dom A ) ) = ran ( A |` ( B i^i dom A ) )
12		df-ima 4782	. . . 4 |- ( A " B ) = ran ( A |` B )
13	10, 11, 12	3eqtr4g 2296	. . 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 4264	. . 3 |- ( B e. C -> ( B i^i dom A ) e. _V )
16		inss2 3452	. . . 4 |- ( B i^i dom A ) C_ dom A
17		funimaexglem 5459	. . . 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 1367	. . 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 2316	1 |- ( ( Fun A /\ B e. C ) -> ( A " B ) e. _V )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   i^i cin 3219   C_ wss 3220   dom cdm 4769   ran crn 4770   |` cres 4771   "cima 4772   Rel wrel 4774   Fun wfun 5366

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-fun 5374

This theorem is referenced by:  funimaex  5461  resfunexg  5927  resfunexgALT  6327  fnexALT  6330  suplocexprlem2b  8071  suplocexprlemlub  8081