Theorem iunxpssiun1 32540

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 4991	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
2	1	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
3		nfcv 2894	. . . . . . 7 ⊢ Ⅎ𝑦𝐵
4		nfcsb1v 3869	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
5		csbeq1a 3859	. . . . . . 7 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
6	3, 4, 5	cbviun 4980	. . . . . 6 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵
7	2, 6	sseqtrdi 3970	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ ∪ 𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵)
8		iunxpssiun1.1	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ⊆ 𝐸)
9		xpss12 5626	. . . . 5 ⊢ ((𝐵 ⊆ ∪ 𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 ∧ 𝐶 ⊆ 𝐸) → (𝐵 × 𝐶) ⊆ (∪ 𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 × 𝐸))
10	7, 8, 9	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 × 𝐶) ⊆ (∪ 𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 × 𝐸))
11	10	ralrimiva 3124	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐵 × 𝐶) ⊆ (∪ 𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 × 𝐸))
12		nfcv 2894	. . . . . 6 ⊢ Ⅎ𝑥𝐴
13	12, 4	nfiun 4968	. . . . 5 ⊢ Ⅎ𝑥∪ 𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵
14		nfcv 2894	. . . . 5 ⊢ Ⅎ𝑥𝐸
15	13, 14	nfxp 5644	. . . 4 ⊢ Ⅎ𝑥(∪ 𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 × 𝐸)
16	15	iunssf 4988	. . 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 234	. 2 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 (𝐵 × 𝐶) ⊆ (∪ 𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 × 𝐸))
18	6	xpeq1i 5637	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 × 𝐸) = (∪ 𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 × 𝐸)
19	17, 18	sseqtrrdi 3971	1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 (𝐵 × 𝐶) ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 × 𝐸))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2111  ∀wral 3047  ⦋csb 3845   ⊆ wss 3897  ∪ ciun 4936   × cxp 5609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-v 3438  df-sbc 3737  df-csb 3846  df-ss 3914  df-iun 4938  df-opab 5149  df-xp 5617

This theorem is referenced by:  fldextrspunlsplem  33678