Theorem mpoexw 6439

Description: Weak version of mpoex 6440 that holds without ax-coll 4241. If the domain and codomain of an operation given by maps-to notation are sets, the operation is a set. (Contributed by Rohan Ridenour, 14-Aug-2023.)

Hypotheses

Ref	Expression
mpoexw.1	|- A e. _V
mpoexw.2	|- B e. _V
mpoexw.3	|- D e. _V
mpoexw.4	|- A. x e. A A. y e. B C e. D

Assertion

Ref	Expression
mpoexw	|- ( x e. A , y e. B |-> C ) e. _V

Distinct variable groups:   x, y, A   x, B, y   x, D, y

Allowed substitution hints:   C( x, y)

Proof of Theorem mpoexw

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2238	. . 3 |- ( x e. A , y e. B |-> C ) = ( x e. A , y e. B |-> C )
2	1	mpofun 6180	. 2 |- Fun ( x e. A , y e. B |-> C )
3		mpoexw.4	. . . 4 |- A. x e. A A. y e. B C e. D
4	1	dmmpoga 6434	. . . 4 |- ( A. x e. A A. y e. B C e. D -> dom ( x e. A , y e. B |-> C ) = ( A X. B ) )
5	3, 4	ax-mp 5	. . 3 |- dom ( x e. A , y e. B |-> C ) = ( A X. B )
6		mpoexw.1	. . . 4 |- A e. _V
7		mpoexw.2	. . . 4 |- B e. _V
8	6, 7	xpex 4886	. . 3 |- ( A X. B ) e. _V
9	5, 8	eqeltri 2311	. 2 |- dom ( x e. A , y e. B |-> C ) e. _V
10	1	rnmpo 6189	. . 3 |- ran ( x e. A , y e. B |-> C ) = { z | E. x e. A E. y e. B z = C }
11		mpoexw.3	. . . 4 |- D e. _V
12	3	rspec 2602	. . . . . . . . 9 |- ( x e. A -> A. y e. B C e. D )
13	12	r19.21bi 2638	. . . . . . . 8 |- ( ( x e. A /\ y e. B ) -> C e. D )
14		eleq1a 2310	. . . . . . . 8 |- ( C e. D -> ( z = C -> z e. D ) )
15	13, 14	syl 14	. . . . . . 7 |- ( ( x e. A /\ y e. B ) -> ( z = C -> z e. D ) )
16	15	rexlimdva 2668	. . . . . 6 |- ( x e. A -> ( E. y e. B z = C -> z e. D ) )
17	16	rexlimiv 2662	. . . . 5 |- ( E. x e. A E. y e. B z = C -> z e. D )
18	17	abssi 3323	. . . 4 |- { z | E. x e. A E. y e. B z = C } C_ D
19	11, 18	ssexi 4266	. . 3 |- { z | E. x e. A E. y e. B z = C } e. _V
20	10, 19	eqeltri 2311	. 2 |- ran ( x e. A , y e. B |-> C ) e. _V
21		funexw 6331	. 2 |- ( ( Fun ( x e. A , y e. B |-> C ) /\ dom ( x e. A , y e. B |-> C ) e. _V /\ ran ( x e. A , y e. B |-> C ) e. _V ) -> ( x e. A , y e. B |-> C ) e. _V )
22	2, 9, 20, 21	mp3an 1378	1 |- ( x e. A , y e. B |-> C ) e. _V

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   {cab 2224   A.wral 2528   E.wrex 2529   _Vcvv 2821   X. cxp 4767   dom cdm 4769   ran crn 4770   Fun wfun 5366   e. cmpo 6077

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-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

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-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  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-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365

This theorem is referenced by:  prdsvallem  13598