Theorem mpoexw 6317

Description: Weak version of mpoex 6318 that holds without ax-coll 4170. 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 2206	. . 3 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
2	1	mpofun 6065	. 2 ⊢ Fun (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
3		mpoexw.4	. . . 4 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷
4	1	dmmpoga 6312	. . . 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 4803	. . 3 ⊢ (𝐴 × 𝐵) ∈ V
9	5, 8	eqeltri 2279	. 2 ⊢ dom (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V
10	1	rnmpo 6074	. . 3 ⊢ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶}
11		mpoexw.3	. . . 4 ⊢ 𝐷 ∈ V
12	3	rspec 2559	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷)
13	12	r19.21bi 2595	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝐷)
14		eleq1a 2278	. . . . . . . 8 ⊢ (𝐶 ∈ 𝐷 → (𝑧 = 𝐶 → 𝑧 ∈ 𝐷))
15	13, 14	syl 14	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑧 = 𝐶 → 𝑧 ∈ 𝐷))
16	15	rexlimdva 2624	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝑧 ∈ 𝐷))
17	16	rexlimiv 2618	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝑧 ∈ 𝐷)
18	17	abssi 3272	. . . 4 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} ⊆ 𝐷
19	11, 18	ssexi 4193	. . 3 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} ∈ V
20	10, 19	eqeltri 2279	. 2 ⊢ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V
21		funexw 6215	. 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 1350	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2177  {cab 2192  ∀wral 2485  ∃wrex 2486  Vcvv 2773   × cxp 4686  dom cdm 4688  ran crn 4689  Fun wfun 5279   ∈ cmpo 5964

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-pow 4229  ax-pr 4264  ax-un 4493

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-id 4353  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-fv 5293  df-oprab 5966  df-mpo 5967  df-1st 6244  df-2nd 6245

This theorem is referenced by:  prdsvallem  13189