Theorem djussxp2 33004

Description: Stronger version of djussxp 5830. (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 2924	. . . 4 ⊢ Ⅎ𝑘𝐴
2		nfiu1 4991	. . . 4 ⊢ Ⅎ𝑘∪ 𝑘 ∈ 𝐴 𝐵
3	1, 2	nfxp 5693	. . 3 ⊢ Ⅎ𝑘(𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵)
4	3	iunssf 5006	. 2 ⊢ (∪ 𝑘 ∈ 𝐴 ({𝑘} × 𝐵) ⊆ (𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵) ↔ ∀𝑘 ∈ 𝐴 ({𝑘} × 𝐵) ⊆ (𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵))
5		snssi 4750	. . 3 ⊢ (𝑘 ∈ 𝐴 → {𝑘} ⊆ 𝐴)
6		ssiun2 5011	. . 3 ⊢ (𝑘 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑘 ∈ 𝐴 𝐵)
7		xpss12 5675	. . 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 3085	1 ⊢ ∪ 𝑘 ∈ 𝐴 ({𝑘} × 𝐵) ⊆ (𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵)

This proof depends on syntax axioms:   ∈ wcel 2142   ⊆ wss 3904  {csn 4588  ∪ ciun 4955   × cxp 5658

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1572  df-ex 1809  df-nf 1813  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rex 3089  df-v 3456  df-ss 3921  df-sn 4589  df-iun 4957  df-opab 5173  df-xp 5666

This theorem is used by:  2ndresdju  33005  gsumpart  33392