Theorem iunxpssiun1 32657

Description: Provide an upper bound for the indexed union of cartesian products. (Contributed by Thierry Arnoux, 13-Oct-2025.)

Hypothesis

Ref	Expression
iunxpssiun1.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ⊆ 𝐸)

Assertion

Ref	Expression
iunxpssiun1	⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 (𝐵 × 𝐶) ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 × 𝐸))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐸   𝜑,𝑥

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem iunxpssiun1

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssiun2 4977	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
2	1	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
3		nfcv 2901	. . . . . . 7 ⊢ Ⅎ𝑦𝐵
4		nfcsb1v 3855	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
5		csbeq1a 3845	. . . . . . 7 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
6	3, 4, 5	cbviun 4964	. . . . . 6 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵
7	2, 6	sseqtrdi 3955	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ ∪ 𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵)
8		iunxpssiun1.1	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ⊆ 𝐸)
9		xpss12 5633	. . . . 5 ⊢ ((𝐵 ⊆ ∪ 𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 ∧ 𝐶 ⊆ 𝐸) → (𝐵 × 𝐶) ⊆ (∪ 𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 × 𝐸))
10	7, 8, 9	syl2anc 590	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 × 𝐶) ⊆ (∪ 𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 × 𝐸))
11	10	ralrimiva 3131	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐵 × 𝐶) ⊆ (∪ 𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 × 𝐸))
12		nfcv 2901	. . . . . 6 ⊢ Ⅎ𝑥𝐴
13	12, 4	nfiun 4953	. . . . 5 ⊢ Ⅎ𝑥∪ 𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵
14		nfcv 2901	. . . . 5 ⊢ Ⅎ𝑥𝐸
15	13, 14	nfxp 5651	. . . 4 ⊢ Ⅎ𝑥(∪ 𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 × 𝐸)
16	15	iunssf 4972	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 (𝐵 × 𝐶) ⊆ (∪ 𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 × 𝐸) ↔ ∀𝑥 ∈ 𝐴 (𝐵 × 𝐶) ⊆ (∪ 𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 × 𝐸))
17	11, 16	sylibr 235	. 2 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 (𝐵 × 𝐶) ⊆ (∪ 𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 × 𝐸))
18	6	xpeq1i 5644	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 × 𝐸) = (∪ 𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 × 𝐸)
19	17, 18	sseqtrrdi 3956	1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 (𝐵 × 𝐶) ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 × 𝐸))

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2119  ∀wral 3053  ⦋csb 3831   ⊆ wss 3883  ∪ ciun 4921   × cxp 5616

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

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ral 3054  df-rex 3064  df-v 3433  df-sbc 3724  df-csb 3832  df-ss 3900  df-iun 4923  df-opab 5135  df-xp 5624

This theorem is referenced by:  fldextrspunlsplem  33857