Theorem ssrel2 5739

Description: A subclass relationship depends only on a relation's ordered pairs. This version of ssrel 5736 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 3935	. . . 4 ⊢ (𝑅 ⊆ 𝑆 → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆))
2	1	a1d 25	. . 3 ⊢ (𝑅 ⊆ 𝑆 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆)))
3	2	ralrimivv 3193	. 2 ⊢ (𝑅 ⊆ 𝑆 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆))
4		eleq1 2825	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
5		eleq1 2825	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑆 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑆))
6	4, 5	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆)))
7	6	biimprcd 249	. . . . . . . . 9 ⊢ ((⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆)))
8	7	2ralimi 3124	. . . . . . . 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 3177	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆)) ↔ (∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆)))
10	9	ralbii 3094	. . . . . . . . 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 3177	. . . . . . . . 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 274	. . . . . . . 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 217	. . . . . . 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 86	. . . . . 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 29	. . . . 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 1919	. . . 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		dfss2 3928	. . . . 5 ⊢ (𝑅 ⊆ (𝐴 × 𝐵) ↔ ∀𝑧(𝑧 ∈ 𝑅 → 𝑧 ∈ (𝐴 × 𝐵)))
18		elxp2 5655	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩)
19	18	imbi2i 335	. . . . . 6 ⊢ ((𝑧 ∈ 𝑅 → 𝑧 ∈ (𝐴 × 𝐵)) ↔ (𝑧 ∈ 𝑅 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩))
20	19	albii 1821	. . . . 5 ⊢ (∀𝑧(𝑧 ∈ 𝑅 → 𝑧 ∈ (𝐴 × 𝐵)) ↔ ∀𝑧(𝑧 ∈ 𝑅 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩))
21	17, 20	bitri 274	. . . 4 ⊢ (𝑅 ⊆ (𝐴 × 𝐵) ↔ ∀𝑧(𝑧 ∈ 𝑅 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩))
22		dfss2 3928	. . . 4 ⊢ (𝑅 ⊆ 𝑆 ↔ ∀𝑧(𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆))
23	16, 21, 22	3imtr4g 295	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → (𝑅 ⊆ (𝐴 × 𝐵) → 𝑅 ⊆ 𝑆))
24	23	com12 32	. 2 ⊢ (𝑅 ⊆ (𝐴 × 𝐵) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → 𝑅 ⊆ 𝑆))
25	3, 24	impbid2 225	1 ⊢ (𝑅 ⊆ (𝐴 × 𝐵) → (𝑅 ⊆ 𝑆 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1539   = wceq 1541   ∈ wcel 2106  ∀wral 3062  ∃wrex 3071   ⊆ wss 3908  ⟨cop 4590   × cxp 5629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3063  df-rex 3072  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-opab 5166  df-xp 5637

This theorem is referenced by:  metuel2  23905  isarchi  31901