Theorem funimaexg 5357

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 5287	. . 3 |- ( Fun A -> Rel A )
3		resres 4970	. . . . . . 7 |- ( ( A |` dom A ) |` B ) = ( A |` ( dom A i^i B ) )
4		incom 3364	. . . . . . . 8 |- ( B i^i dom A ) = ( dom A i^i B )
5	4	reseq2i 4955	. . . . . . 7 |- ( A |` ( B i^i dom A ) ) = ( A |` ( dom A i^i B ) )
6	3, 5	eqtr4i 2228	. . . . . 6 |- ( ( A |` dom A ) |` B ) = ( A |` ( B i^i dom A ) )
7		resdm 4997	. . . . . . 7 |- ( Rel A -> ( A |` dom A ) = A )
8	7	reseq1d 4957	. . . . . 6 |- ( Rel A -> ( ( A |` dom A ) |` B ) = ( A |` B ) )
9	6, 8	eqtr3id 2251	. . . . 5 |- ( Rel A -> ( A |` ( B i^i dom A ) ) = ( A |` B ) )
10	9	rneqd 4906	. . . 4 |- ( Rel A -> ran ( A |` ( B i^i dom A ) ) = ran ( A |` B ) )
11		df-ima 4687	. . . 4 |- ( A " ( B i^i dom A ) ) = ran ( A |` ( B i^i dom A ) )
12		df-ima 4687	. . . 4 |- ( A " B ) = ran ( A |` B )
13	10, 11, 12	3eqtr4g 2262	. . 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 4179	. . 3 |- ( B e. C -> ( B i^i dom A ) e. _V )
16		inss2 3393	. . . 4 |- ( B i^i dom A ) C_ dom A
17		funimaexglem 5356	. . . 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 1338	. . 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 2282	1 |- ( ( Fun A /\ B e. C ) -> ( A " B ) e. _V )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1372   e. wcel 2175   _Vcvv 2771   i^i cin 3164   C_ wss 3165   dom cdm 4674   ran crn 4675   |` cres 4676   "cima 4677   Rel wrel 4679   Fun wfun 5264

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-fun 5272

This theorem is referenced by:  funimaex  5358  resfunexg  5804  resfunexgALT  6192  fnexALT  6195  suplocexprlem2b  7826  suplocexprlemlub  7836