Theorem ovmptss 8109

Description: If all the values of the mapping are subsets of a class 𝑋, then so is any evaluation of the mapping. (Contributed by Mario Carneiro, 24-Dec-2016.) Avoid ax-un 7755 and shorten proof. (Revised by Zhi Wang, 16-Sep-2026.)

Hypothesis

Ref	Expression
ovmptss.1	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Assertion

Ref	Expression
ovmptss	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ⊆ 𝑋 → (𝐸𝐹𝐺) ⊆ 𝑋)

Distinct variable groups:   𝑥,𝑦,𝑋   𝑦,𝐴

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥, 𝑦)   𝐶(𝑥, 𝑦)   𝐸(𝑥, 𝑦)   𝐹(𝑥, 𝑦)   𝐺(𝑥, 𝑦)

Proof of Theorem ovmptss

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovssunirn 7460	. . 3 ⊢ (𝐸(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)𝐺) ⊆ ∪ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
2		sseq1 3934	. . . . . . . . . 10 ⊢ (𝑧 = 𝐶 → (𝑧 ⊆ 𝑋 ↔ 𝐶 ⊆ 𝑋))
3	2	biimprd 251	. . . . . . . . 9 ⊢ (𝑧 = 𝐶 → (𝐶 ⊆ 𝑋 → 𝑧 ⊆ 𝑋))
4	3	com12 33	. . . . . . . 8 ⊢ (𝐶 ⊆ 𝑋 → (𝑧 = 𝐶 → 𝑧 ⊆ 𝑋))
5	4	2ralimi 3132	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ⊆ 𝑋 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑧 = 𝐶 → 𝑧 ⊆ 𝑋))
6		r19.23v 3189	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐵 (𝑧 = 𝐶 → 𝑧 ⊆ 𝑋) ↔ (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝑧 ⊆ 𝑋))
7	6	ralbii 3108	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑧 = 𝐶 → 𝑧 ⊆ 𝑋) ↔ ∀𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝑧 ⊆ 𝑋))
8		r19.23v 3189	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝑧 ⊆ 𝑋) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝑧 ⊆ 𝑋))
9	7, 8	bitri 278	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑧 = 𝐶 → 𝑧 ⊆ 𝑋) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝑧 ⊆ 𝑋))
10	5, 9	sylib 221	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ⊆ 𝑋 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝑧 ⊆ 𝑋))
11		eqid 2760	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
12	11	rnmpo 7557	. . . . . . . . 9 ⊢ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶}
13	12	eleq2i 2852	. . . . . . . 8 ⊢ (𝑧 ∈ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ↔ 𝑧 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶})
14		abid 2742	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶)
15	13, 14	bitri 278	. . . . . . 7 ⊢ (𝑧 ∈ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶)
16	15	imbi1i 352	. . . . . 6 ⊢ ((𝑧 ∈ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) → 𝑧 ⊆ 𝑋) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝑧 ⊆ 𝑋))
17	10, 16	sylibr 237	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ⊆ 𝑋 → (𝑧 ∈ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) → 𝑧 ⊆ 𝑋))
18	17	ralrimiv 3153	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ⊆ 𝑋 → ∀𝑧 ∈ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑧 ⊆ 𝑋)
19		unissb 4900	. . . 4 ⊢ (∪ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⊆ 𝑋 ↔ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑧 ⊆ 𝑋)
20	18, 19	sylibr 237	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ⊆ 𝑋 → ∪ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⊆ 𝑋)
21	1, 20	sstrid 3920	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ⊆ 𝑋 → (𝐸(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)𝐺) ⊆ 𝑋)
22		ovmptss.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
23	22	oveqi 7437	. . 3 ⊢ (𝐸𝐹𝐺) = (𝐸(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)𝐺)
24	23	sseq1i 3937	. 2 ⊢ ((𝐸𝐹𝐺) ⊆ 𝑋 ↔ (𝐸(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)𝐺) ⊆ 𝑋)
25	21, 24	sylibr 237	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ⊆ 𝑋 → (𝐸𝐹𝐺) ⊆ 𝑋)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  {cab 2738  ∀wral 3076  ∃wrex 3086   ⊆ wss 3896  ∪ cuni 4866  ran crn 5652  (class class class)co 7424   ∈ cmpo 7426

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-cnv 5659  df-dm 5661  df-rn 5662  df-iota 6497  df-fv 6551  df-ov 7427  df-oprab 7428  df-mpo 7429

This theorem is used by:  relmpoopab  8111  relxpchom  18364  reldv  26199  parhomssun  50883