Theorem iunxpssiun1 33155

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 5006	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
2	1	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
3		nfcv 2923	. . . . . . 7 ⊢ Ⅎ𝑦𝐵
4		nfcsb1v 3871	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
5		csbeq1a 3861	. . . . . . 7 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
6	3, 4, 5	cbviun 4993	. . . . . 6 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵
7	2, 6	sseqtrdi 3971	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ ∪ 𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵)
8		iunxpssiun1.1	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ⊆ 𝐸)
9		xpss12 5666	. . . . 5 ⊢ ((𝐵 ⊆ ∪ 𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 ∧ 𝐶 ⊆ 𝐸) → (𝐵 × 𝐶) ⊆ (∪ 𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 × 𝐸))
10	7, 8, 9	syl2anc 596	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 × 𝐶) ⊆ (∪ 𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 × 𝐸))
11	10	ralrimiva 3155	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐵 × 𝐶) ⊆ (∪ 𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 × 𝐸))
12		nfcv 2923	. . . . . 6 ⊢ Ⅎ𝑥𝐴
13	12, 4	nfiun 4982	. . . . 5 ⊢ Ⅎ𝑥∪ 𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵
14		nfcv 2923	. . . . 5 ⊢ Ⅎ𝑥𝐸
15	13, 14	nfxp 5684	. . . 4 ⊢ Ⅎ𝑥(∪ 𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 × 𝐸)
16	15	iunssf 5001	. . 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 237	. 2 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 (𝐵 × 𝐶) ⊆ (∪ 𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 × 𝐸))
18	6	xpeq1i 5677	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 × 𝐸) = (∪ 𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 × 𝐸)
19	17, 18	sseqtrrdi 3972	1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 (𝐵 × 𝐶) ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 × 𝐸))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145  ∀wral 3077  ⦋csb 3847   ⊆ wss 3899  ∪ ciun 4951   × cxp 5649

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-v 3453  df-sbc 3740  df-csb 3848  df-ss 3916  df-iun 4953  df-opab 5168  df-xp 5657

This theorem is used by:  fldextrspunlsplem  34298