Theorem djussxp2 32665

Description: Stronger version of djussxp 5859. (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 2903	. . . 4 ⊢ Ⅎ𝑘𝐴
2		nfiu1 5032	. . . 4 ⊢ Ⅎ𝑘∪ 𝑘 ∈ 𝐴 𝐵
3	1, 2	nfxp 5722	. . 3 ⊢ Ⅎ𝑘(𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵)
4	3	iunssf 5049	. 2 ⊢ (∪ 𝑘 ∈ 𝐴 ({𝑘} × 𝐵) ⊆ (𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵) ↔ ∀𝑘 ∈ 𝐴 ({𝑘} × 𝐵) ⊆ (𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵))
5		snssi 4813	. . 3 ⊢ (𝑘 ∈ 𝐴 → {𝑘} ⊆ 𝐴)
6		ssiun2 5052	. . 3 ⊢ (𝑘 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑘 ∈ 𝐴 𝐵)
7		xpss12 5704	. . 3 ⊢ (({𝑘} ⊆ 𝐴 ∧ 𝐵 ⊆ ∪ 𝑘 ∈ 𝐴 𝐵) → ({𝑘} × 𝐵) ⊆ (𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵))
8	5, 6, 7	syl2anc 584	. 2 ⊢ (𝑘 ∈ 𝐴 → ({𝑘} × 𝐵) ⊆ (𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵))
9	4, 8	mprgbir 3066	1 ⊢ ∪ 𝑘 ∈ 𝐴 ({𝑘} × 𝐵) ⊆ (𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵)

Syntax hints:   ∈ wcel 2106   ⊆ wss 3963  {csn 4631  ∪ ciun 4996   × cxp 5687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-v 3480  df-ss 3980  df-sn 4632  df-iun 4998  df-opab 5211  df-xp 5695

This theorem is referenced by:  2ndresdju  32666  gsumpart  33043