Theorem mpoexw 8020

Description: Weak version of mpoex 8021 that holds without ax-rep 5199. 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 2739	. . 3 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
2	1	mpofun 7480	. 2 ⊢ Fun (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
3		mpoexw.4	. . . 4 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷
4	1	dmmpoga 8015	. . . 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 7696	. . 3 ⊢ (𝐴 × 𝐵) ∈ V
9	5, 8	eqeltri 2835	. 2 ⊢ dom (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V
10	1	rnmpo 7489	. . 3 ⊢ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶}
11		mpoexw.3	. . . 4 ⊢ 𝐷 ∈ V
12	3	rspec 3230	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷)
13	12	r19.21bi 3231	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝐷)
14		eleq1a 2834	. . . . . . . 8 ⊢ (𝐶 ∈ 𝐷 → (𝑧 = 𝐶 → 𝑧 ∈ 𝐷))
15	13, 14	syl 17	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑧 = 𝐶 → 𝑧 ∈ 𝐷))
16	15	rexlimdva 3140	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝑧 ∈ 𝐷))
17	16	rexlimiv 3133	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝑧 ∈ 𝐷)
18	17	abssi 3999	. . . 4 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} ⊆ 𝐷
19	11, 18	ssexi 5250	. . 3 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} ∈ V
20	10, 19	eqeltri 2835	. 2 ⊢ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V
21		funexw 7894	. 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 1469	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {cab 2717  ∀wral 3053  ∃wrex 3063  Vcvv 3431   × cxp 5616  dom cdm 5618  ran crn 5619  Fun wfun 6479   ∈ cmpo 7358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-fv 6493  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932

This theorem is referenced by:  mptmpoopabbrd  8022  prdsvallem  17408  prdsds  17418  plusffval  18605  grpsubfval  18950  mulgfval  19036  scaffval  20870  ipffval  21623