Theorem bj-ssrel 38168

Description: Variant of ssrel 5765 where the antecedent is more versatile than Rel 𝐴 (the proof is shorter than that of ssrel 5765, which can be proved from this one using sylbi 220 and df-rel 5663, and of ssrel2 5768, which uses restricted quantifiers). (Contributed by BJ, 7-Nov-2026.)

Assertion

Ref	Expression
bj-ssrel	⊢ (𝐴 ⊆ (𝑉 × 𝑊) → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑉,𝑦   𝑥,𝑊,𝑦

Proof of Theorem bj-ssrel

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3907	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
2	1	alrimivv 1963	. 2 ⊢ (𝐴 ⊆ 𝐵 → ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
3		ssel 3907	. . . . . . 7 ⊢ (𝐴 ⊆ (𝑉 × 𝑊) → (𝑧 ∈ 𝐴 → 𝑧 ∈ (𝑉 × 𝑊)))
4		elxpi 5678	. . . . . . 7 ⊢ (𝑧 ∈ (𝑉 × 𝑊) → ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑊)))
5	3, 4	syl6 36	. . . . . 6 ⊢ (𝐴 ⊆ (𝑉 × 𝑊) → (𝑧 ∈ 𝐴 → ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑊))))
6		eleq1 2852	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
7		eleq1 2852	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝐵 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
8	6, 7	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
9	8	adantr 486	. . . . . . . . 9 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑊)) → ((𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
10	9	biimprcd 253	. . . . . . . 8 ⊢ ((⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑊)) → (𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵)))
11	10	2alimi 1845	. . . . . . 7 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → ∀𝑥∀𝑦((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑊)) → (𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵)))
12		19.23vv 1978	. . . . . . 7 ⊢ (∀𝑥∀𝑦((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑊)) → (𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵)) ↔ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑊)) → (𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵)))
13	11, 12	sylib 221	. . . . . 6 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑊)) → (𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵)))
14	5, 13	sylan9 517	. . . . 5 ⊢ ((𝐴 ⊆ (𝑉 × 𝑊) ∧ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)) → (𝑧 ∈ 𝐴 → (𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵)))
15	14	pm2.43d 54	. . . 4 ⊢ ((𝐴 ⊆ (𝑉 × 𝑊) ∧ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)) → (𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵))
16	15	ssrdv 3919	. . 3 ⊢ ((𝐴 ⊆ (𝑉 × 𝑊) ∧ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)) → 𝐴 ⊆ 𝐵)
17	16	ex 418	. 2 ⊢ (𝐴 ⊆ (𝑉 × 𝑊) → (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → 𝐴 ⊆ 𝐵))
18	2, 17	impbid2 229	1 ⊢ (𝐴 ⊆ (𝑉 × 𝑊) → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2147   ⊆ wss 3900  ⟨cop 4838   × cxp 5654

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ss 3901  df-opab 5172  df-xp 5662

This theorem is used by: (None)