Theorem ovmptss 8139

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 7783 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 7487	. . 3 ⊢ (𝐸(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)𝐺) ⊆ ∪ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
2		sseq1 3938	. . . . . . . . . 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 3136	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ⊆ 𝑋 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑧 = 𝐶 → 𝑧 ⊆ 𝑋))
6		r19.23v 3193	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐵 (𝑧 = 𝐶 → 𝑧 ⊆ 𝑋) ↔ (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝑧 ⊆ 𝑋))
7	6	ralbii 3112	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑧 = 𝐶 → 𝑧 ⊆ 𝑋) ↔ ∀𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = 𝐶 → 𝑧 ⊆ 𝑋))
8		r19.23v 3193	. . . . . . . 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 2763	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
12	11	rnmpo 7585	. . . . . . . . 9 ⊢ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶}
13	12	eleq2i 2856	. . . . . . . 8 ⊢ (𝑧 ∈ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ↔ 𝑧 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶})
14		abid 2745	. . . . . . . 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 3157	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ⊆ 𝑋 → ∀𝑧 ∈ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑧 ⊆ 𝑋)
19		unissb 4906	. . . 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 3924	. 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 7464	. . 3 ⊢ (𝐸𝐹𝐺) = (𝐸(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)𝐺)
24	23	sseq1i 3941	. 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 1570   ∈ wcel 2147  {cab 2741  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900  ∪ cuni 4872  ran crn 5659  (class class class)co 7451   ∈ cmpo 7453

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-cnv 5666  df-dm 5668  df-rn 5669  df-iota 6520  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456

This theorem is used by:  relmpoopab  8141  relxpchom  18405  reldv  26245  parhomssun  51078