Theorem djussxp2 32630

Description: Stronger version of djussxp 5784. (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 2894	. . . 4 ⊢ Ⅎ𝑘𝐴
2		nfiu1 4975	. . . 4 ⊢ Ⅎ𝑘∪ 𝑘 ∈ 𝐴 𝐵
3	1, 2	nfxp 5647	. . 3 ⊢ Ⅎ𝑘(𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵)
4	3	iunssf 4991	. 2 ⊢ (∪ 𝑘 ∈ 𝐴 ({𝑘} × 𝐵) ⊆ (𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵) ↔ ∀𝑘 ∈ 𝐴 ({𝑘} × 𝐵) ⊆ (𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵))
5		snssi 4757	. . 3 ⊢ (𝑘 ∈ 𝐴 → {𝑘} ⊆ 𝐴)
6		ssiun2 4994	. . 3 ⊢ (𝑘 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑘 ∈ 𝐴 𝐵)
7		xpss12 5629	. . 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 3054	1 ⊢ ∪ 𝑘 ∈ 𝐴 ({𝑘} × 𝐵) ⊆ (𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵)

Syntax hints:   ∈ wcel 2111   ⊆ wss 3897  {csn 4573  ∪ ciun 4939   × cxp 5612

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-ss 3914  df-sn 4574  df-iun 4941  df-opab 5152  df-xp 5620

This theorem is referenced by:  2ndresdju  32631  gsumpart  33037