Theorem djussxp2

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

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

Proof

Step	Hyp	Ref	Expression
1		nfcv	⊢ Ⅎ 𝑘 𝐴
2		nfiu1	⊢ Ⅎ 𝑘 ∪ 𝑘 ∈ 𝐴 𝐵
3	1 2	nfxp	⊢ Ⅎ 𝑘 ( 𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵 )
4	3	iunssf	⊢ ( ∪ 𝑘 ∈ 𝐴 ( { 𝑘 } × 𝐵 ) ⊆ ( 𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵 ) ↔ ∀ 𝑘 ∈ 𝐴 ( { 𝑘 } × 𝐵 ) ⊆ ( 𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵 ) )
5		snssi	⊢ ( 𝑘 ∈ 𝐴 → { 𝑘 } ⊆ 𝐴 )
6		ssiun2	⊢ ( 𝑘 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑘 ∈ 𝐴 𝐵 )
7		xpss12	⊢ ( ( { 𝑘 } ⊆ 𝐴 ∧ 𝐵 ⊆ ∪ 𝑘 ∈ 𝐴 𝐵 ) → ( { 𝑘 } × 𝐵 ) ⊆ ( 𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵 ) )
8	5 6 7	syl2anc	⊢ ( 𝑘 ∈ 𝐴 → ( { 𝑘 } × 𝐵 ) ⊆ ( 𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵 ) )
9	4 8	mprgbir	⊢ ∪ 𝑘 ∈ 𝐴 ( { 𝑘 } × 𝐵 ) ⊆ ( 𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵 )