Theorem mpoexw 6357

Description: Weak version of mpoex 6358 that holds without ax-coll 4198. 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 2229	. . 3 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
2	1	mpofun 6105	. 2 ⊢ Fun (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
3		mpoexw.4	. . . 4 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷
4	1	dmmpoga 6352	. . . 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 4833	. . 3 ⊢ (𝐴 × 𝐵) ∈ V
9	5, 8	eqeltri 2302	. 2 ⊢ dom (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V
10	1	rnmpo 6114	. . 3 ⊢ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶}
11		mpoexw.3	. . . 4 ⊢ 𝐷 ∈ V
12	3	rspec 2582	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷)
13	12	r19.21bi 2618	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝐷)
14		eleq1a 2301	. . . . . . . 8 ⊢ (𝐶 ∈ 𝐷 → (𝑧 = 𝐶 → 𝑧 ∈ 𝐷))
15	13, 14	syl 14	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑧 = 𝐶 → 𝑧 ∈ 𝐷))
16	15	rexlimdva 2648	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝑧 ∈ 𝐷))
17	16	rexlimiv 2642	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝑧 ∈ 𝐷)
18	17	abssi 3299	. . . 4 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} ⊆ 𝐷
19	11, 18	ssexi 4221	. . 3 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} ∈ V
20	10, 19	eqeltri 2302	. 2 ⊢ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V
21		funexw 6255	. 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 1371	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200  {cab 2215  ∀wral 2508  ∃wrex 2509  Vcvv 2799   × cxp 4716  dom cdm 4718  ran crn 4719  Fun wfun 5311   ∈ cmpo 6002

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-fv 5325  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285

This theorem is referenced by:  prdsvallem  13300