Theorem ssrel2 5760

Description: A subclass relationship depends only on a relation's ordered pairs. This version of ssrel 5758 is restricted to the relation's domain. (Contributed by Thierry Arnoux, 25-Jan-2018.)

Assertion

Ref	Expression
ssrel2	⊢ (𝑅 ⊆ (𝐴 × 𝐵) → (𝑅 ⊆ 𝑆 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆)))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦

Proof of Theorem ssrel2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3902	. . . 4 ⊢ (𝑅 ⊆ 𝑆 → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆))
2	1	a1d 26	. . 3 ⊢ (𝑅 ⊆ 𝑆 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆)))
3	2	ralrimivv 3203	. 2 ⊢ (𝑅 ⊆ 𝑆 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆))
4		eleq1 2848	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
5		eleq1 2848	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑆 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑆))
6	4, 5	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆)))
7	6	biimprcd 253	. . . . . . . . 9 ⊢ ((⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆)))
8	7	2ralimi 3132	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆)))
9		r19.23v 3189	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆)) ↔ (∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆)))
10	9	ralbii 3108	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆)) ↔ ∀𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆)))
11		r19.23v 3189	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆)) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆)))
12	10, 11	bitri 278	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆)) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆)))
13	8, 12	sylib 221	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆)))
14	13	com23 87	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → (𝑧 ∈ 𝑅 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 ∈ 𝑆)))
15	14	a2d 30	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → ((𝑧 ∈ 𝑅 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩) → (𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆)))
16	15	alimdv 1948	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → (∀𝑧(𝑧 ∈ 𝑅 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩) → ∀𝑧(𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆)))
17		df-ss 3896	. . . . 5 ⊢ (𝑅 ⊆ (𝐴 × 𝐵) ↔ ∀𝑧(𝑧 ∈ 𝑅 → 𝑧 ∈ (𝐴 × 𝐵)))
18		elxp2 5674	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩)
19	18	imbi2i 339	. . . . . 6 ⊢ ((𝑧 ∈ 𝑅 → 𝑧 ∈ (𝐴 × 𝐵)) ↔ (𝑧 ∈ 𝑅 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩))
20	19	albii 1851	. . . . 5 ⊢ (∀𝑧(𝑧 ∈ 𝑅 → 𝑧 ∈ (𝐴 × 𝐵)) ↔ ∀𝑧(𝑧 ∈ 𝑅 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩))
21	17, 20	bitri 278	. . . 4 ⊢ (𝑅 ⊆ (𝐴 × 𝐵) ↔ ∀𝑧(𝑧 ∈ 𝑅 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩))
22		df-ss 3896	. . . 4 ⊢ (𝑅 ⊆ 𝑆 ↔ ∀𝑧(𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆))
23	16, 21, 22	3imtr4g 299	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → (𝑅 ⊆ (𝐴 × 𝐵) → 𝑅 ⊆ 𝑆))
24	23	com12 33	. 2 ⊢ (𝑅 ⊆ (𝐴 × 𝐵) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → 𝑅 ⊆ 𝑆))
25	3, 24	impbid2 229	1 ⊢ (𝑅 ⊆ (𝐴 × 𝐵) → (𝑅 ⊆ 𝑆 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1567   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086   ⊆ wss 3895  ⟨cop 4834   × cxp 5648

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-in 4091  df-un 4151  df-sn 4585  df-pr 4587  df-op 4835  df-opab 5167  df-xp 5656

This theorem is used by:  metuel2  24889  isarchi  33789  j0ltalephlem  35946