Theorem djussxp2 32993

Description: Stronger version of djussxp 5831. (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 2925	. . . 4 ⊢ Ⅎ𝑘𝐴
2		nfiu1 4992	. . . 4 ⊢ Ⅎ𝑘∪ 𝑘 ∈ 𝐴 𝐵
3	1, 2	nfxp 5694	. . 3 ⊢ Ⅎ𝑘(𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵)
4	3	iunssf 5007	. 2 ⊢ (∪ 𝑘 ∈ 𝐴 ({𝑘} × 𝐵) ⊆ (𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵) ↔ ∀𝑘 ∈ 𝐴 ({𝑘} × 𝐵) ⊆ (𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵))
5		snssi 4751	. . 3 ⊢ (𝑘 ∈ 𝐴 → {𝑘} ⊆ 𝐴)
6		ssiun2 5012	. . 3 ⊢ (𝑘 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑘 ∈ 𝐴 𝐵)
7		xpss12 5676	. . 3 ⊢ (({𝑘} ⊆ 𝐴 ∧ 𝐵 ⊆ ∪ 𝑘 ∈ 𝐴 𝐵) → ({𝑘} × 𝐵) ⊆ (𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵))
8	5, 6, 7	syl2anc 595	. 2 ⊢ (𝑘 ∈ 𝐴 → ({𝑘} × 𝐵) ⊆ (𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵))
9	4, 8	mprgbir 3086	1 ⊢ ∪ 𝑘 ∈ 𝐴 ({𝑘} × 𝐵) ⊆ (𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵)

Syntax hints:   ∈ wcel 2143   ⊆ wss 3905  {csn 4589  ∪ ciun 4956   × cxp 5659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-v 3457  df-ss 3922  df-sn 4590  df-iun 4958  df-opab 5174  df-xp 5667

This theorem is referenced by:  2ndresdju  32994  gsumpart  33383