Theorem mpoexw 6449

Description: Weak version of mpoex 6450 that holds without ax-coll 4246. 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 ∈ V
mpoexw.2	⊢ 𝐵 ∈ V
mpoexw.3	⊢ 𝐷 ∈ V
mpoexw.4	⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷

Assertion

Ref	Expression
mpoexw	⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ 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 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
2	1	mpofun 6190	. 2 ⊢ Fun (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
3		mpoexw.4	. . . 4 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷
4	1	dmmpoga 6444	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 → dom (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝐴 × 𝐵))
5	3, 4	ax-mp 5	. . 3 ⊢ dom (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝐴 × 𝐵)
6		mpoexw.1	. . . 4 ⊢ 𝐴 ∈ V
7		mpoexw.2	. . . 4 ⊢ 𝐵 ∈ V
8	6, 7	xpex 4891	. . 3 ⊢ (𝐴 × 𝐵) ∈ V
9	5, 8	eqeltri 2311	. 2 ⊢ dom (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V
10	1	rnmpo 6199	. . 3 ⊢ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶}
11		mpoexw.3	. . . 4 ⊢ 𝐷 ∈ V
12	3	rspec 2602	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷)
13	12	r19.21bi 2638	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝐷)
14		eleq1a 2310	. . . . . . . 8 ⊢ (𝐶 ∈ 𝐷 → (𝑧 = 𝐶 → 𝑧 ∈ 𝐷))
15	13, 14	syl 14	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑧 = 𝐶 → 𝑧 ∈ 𝐷))
16	15	rexlimdva 2668	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝑧 ∈ 𝐷))
17	16	rexlimiv 2662	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝑧 ∈ 𝐷)
18	17	abssi 3323	. . . 4 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} ⊆ 𝐷
19	11, 18	ssexi 4271	. . 3 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} ∈ V
20	10, 19	eqeltri 2311	. 2 ⊢ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V
21		funexw 6341	. 2 ⊢ ((Fun (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∧ dom (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V ∧ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V)
22	2, 9, 20, 21	mp3an 1378	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  {cab 2224  ∀wral 2528  ∃wrex 2529  Vcvv 2821   × cxp 4772  dom cdm 4774  ran crn 4775  Fun wfun 5371   ∈ cmpo 6087

This proof depends on 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 4249  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof 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 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375

This theorem is used by:  prdsvallem  13621