Theorem mpoexw 6377

Description: Weak version of mpoex 6378 that holds without ax-coll 4204. 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 2231	. . 3 |- ( x e. A , y e. B |-> C ) = ( x e. A , y e. B |-> C )
2	1	mpofun 6122	. 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 6372	. . . 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 4842	. . 3 |- ( A X. B ) e. _V
9	5, 8	eqeltri 2304	. 2 |- dom ( x e. A , y e. B |-> C ) e. _V
10	1	rnmpo 6131	. . 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 2584	. . . . . . . . 9 |- ( x e. A -> A. y e. B C e. D )
13	12	r19.21bi 2620	. . . . . . . 8 |- ( ( x e. A /\ y e. B ) -> C e. D )
14		eleq1a 2303	. . . . . . . 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 2650	. . . . . 6 |- ( x e. A -> ( E. y e. B z = C -> z e. D ) )
17	16	rexlimiv 2644	. . . . 5 |- ( E. x e. A E. y e. B z = C -> z e. D )
18	17	abssi 3302	. . . 4 |- { z | E. x e. A E. y e. B z = C } C_ D
19	11, 18	ssexi 4227	. . 3 |- { z | E. x e. A E. y e. B z = C } e. _V
20	10, 19	eqeltri 2304	. 2 |- ran ( x e. A , y e. B |-> C ) e. _V
21		funexw 6273	. 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 1373	1 |- ( x e. A , y e. B |-> C ) e. _V

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   {cab 2217   A.wral 2510   E.wrex 2511   _Vcvv 2802   X. cxp 4723   dom cdm 4725   ran crn 4726   Fun wfun 5320   e. cmpo 6019

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303

This theorem is referenced by:  prdsvallem  13354