Theorem funimaexg 5338

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 5271	. . 3 |- ( Fun A -> Rel A )
3		resres 4954	. . . . . . 7 |- ( ( A |` dom A ) |` B ) = ( A |` ( dom A i^i B ) )
4		incom 3351	. . . . . . . 8 |- ( B i^i dom A ) = ( dom A i^i B )
5	4	reseq2i 4939	. . . . . . 7 |- ( A |` ( B i^i dom A ) ) = ( A |` ( dom A i^i B ) )
6	3, 5	eqtr4i 2217	. . . . . 6 |- ( ( A |` dom A ) |` B ) = ( A |` ( B i^i dom A ) )
7		resdm 4981	. . . . . . 7 |- ( Rel A -> ( A |` dom A ) = A )
8	7	reseq1d 4941	. . . . . 6 |- ( Rel A -> ( ( A |` dom A ) |` B ) = ( A |` B ) )
9	6, 8	eqtr3id 2240	. . . . 5 |- ( Rel A -> ( A |` ( B i^i dom A ) ) = ( A |` B ) )
10	9	rneqd 4891	. . . 4 |- ( Rel A -> ran ( A |` ( B i^i dom A ) ) = ran ( A |` B ) )
11		df-ima 4672	. . . 4 |- ( A " ( B i^i dom A ) ) = ran ( A |` ( B i^i dom A ) )
12		df-ima 4672	. . . 4 |- ( A " B ) = ran ( A |` B )
13	10, 11, 12	3eqtr4g 2251	. . 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 4165	. . 3 |- ( B e. C -> ( B i^i dom A ) e. _V )
16		inss2 3380	. . . 4 |- ( B i^i dom A ) C_ dom A
17		funimaexglem 5337	. . . 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 1337	. . 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 2271	1 |- ( ( Fun A /\ B e. C ) -> ( A " B ) e. _V )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   _Vcvv 2760   i^i cin 3152   C_ wss 3153   dom cdm 4659   ran crn 4660   |` cres 4661   "cima 4662   Rel wrel 4664   Fun wfun 5248

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-fun 5256

This theorem is referenced by:  funimaex  5339  resfunexg  5779  resfunexgALT  6160  fnexALT  6163  suplocexprlem2b  7774  suplocexprlemlub  7784