Theorem djussxp2 32666

Description: Stronger version of djussxp 5870. (Contributed by Thierry Arnoux, 23-Jun-2024.)

Assertion

Ref	Expression
djussxp2	⊢ ∪ 𝑘 ∈ 𝐴 ({𝑘} × 𝐵) ⊆ (𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵)

Distinct variable group:   𝐴,𝑘

Allowed substitution hint:   𝐵(𝑘)

Proof of Theorem djussxp2

Step	Hyp	Ref	Expression
1		nfcv 2908	. . . 4 ⊢ Ⅎ𝑘𝐴
2		nfiu1 5050	. . . 4 ⊢ Ⅎ𝑘∪ 𝑘 ∈ 𝐴 𝐵
3	1, 2	nfxp 5733	. . 3 ⊢ Ⅎ𝑘(𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵)
4	3	iunssf 5067	. 2 ⊢ (∪ 𝑘 ∈ 𝐴 ({𝑘} × 𝐵) ⊆ (𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵) ↔ ∀𝑘 ∈ 𝐴 ({𝑘} × 𝐵) ⊆ (𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵))
5		snssi 4833	. . 3 ⊢ (𝑘 ∈ 𝐴 → {𝑘} ⊆ 𝐴)
6		ssiun2 5070	. . 3 ⊢ (𝑘 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑘 ∈ 𝐴 𝐵)
7		xpss12 5715	. . 3 ⊢ (({𝑘} ⊆ 𝐴 ∧ 𝐵 ⊆ ∪ 𝑘 ∈ 𝐴 𝐵) → ({𝑘} × 𝐵) ⊆ (𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵))
8	5, 6, 7	syl2anc 583	. 2 ⊢ (𝑘 ∈ 𝐴 → ({𝑘} × 𝐵) ⊆ (𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵))
9	4, 8	mprgbir 3074	1 ⊢ ∪ 𝑘 ∈ 𝐴 ({𝑘} × 𝐵) ⊆ (𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵)

Syntax hints:   ∈ wcel 2108   ⊆ wss 3976  {csn 4648  ∪ ciun 5015   × cxp 5698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-v 3490  df-ss 3993  df-sn 4649  df-iun 5017  df-opab 5229  df-xp 5706

This theorem is referenced by:  2ndresdju  32667  gsumpart  33038