Theorem ssrel2 4567

Description: A subclass relationship depends only on a relation's ordered pairs. This version of ssrel 4565 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 3041	. . . 4 ⊢ (𝑅 ⊆ 𝑆 → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆))
2	1	a1d 22	. . 3 ⊢ (𝑅 ⊆ 𝑆 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆)))
3	2	ralrimivv 2472	. 2 ⊢ (𝑅 ⊆ 𝑆 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆))
4		eleq1 2162	. . . . . . . . . . . 12 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
5		eleq1 2162	. . . . . . . . . . . 12 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑆 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑆))
6	4, 5	imbi12d 233	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆)))
7	6	biimprcd 159	. . . . . . . . . 10 ⊢ ((⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆)))
8	7	ralimi 2454	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → ∀𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆)))
9	8	ralimi 2454	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆)))
10		r19.23v 2500	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆)) ↔ (∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆)))
11	10	ralbii 2400	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆)) ↔ ∀𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆)))
12		r19.23v 2500	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆)) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆)))
13	11, 12	bitri 183	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆)) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆)))
14	9, 13	sylib 121	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆)))
15	14	com23 78	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → (𝑧 ∈ 𝑅 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 ∈ 𝑆)))
16	15	a2d 26	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → ((𝑧 ∈ 𝑅 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩) → (𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆)))
17	16	alimdv 1818	. . . 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)))
18		dfss2 3036	. . . . 5 ⊢ (𝑅 ⊆ (𝐴 × 𝐵) ↔ ∀𝑧(𝑧 ∈ 𝑅 → 𝑧 ∈ (𝐴 × 𝐵)))
19		elxp2 4495	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩)
20	19	imbi2i 225	. . . . . 6 ⊢ ((𝑧 ∈ 𝑅 → 𝑧 ∈ (𝐴 × 𝐵)) ↔ (𝑧 ∈ 𝑅 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩))
21	20	albii 1414	. . . . 5 ⊢ (∀𝑧(𝑧 ∈ 𝑅 → 𝑧 ∈ (𝐴 × 𝐵)) ↔ ∀𝑧(𝑧 ∈ 𝑅 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩))
22	18, 21	bitri 183	. . . 4 ⊢ (𝑅 ⊆ (𝐴 × 𝐵) ↔ ∀𝑧(𝑧 ∈ 𝑅 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩))
23		dfss2 3036	. . . 4 ⊢ (𝑅 ⊆ 𝑆 ↔ ∀𝑧(𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆))
24	17, 22, 23	3imtr4g 204	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → (𝑅 ⊆ (𝐴 × 𝐵) → 𝑅 ⊆ 𝑆))
25	24	com12 30	. 2 ⊢ (𝑅 ⊆ (𝐴 × 𝐵) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → 𝑅 ⊆ 𝑆))
26	3, 25	impbid2 142	1 ⊢ (𝑅 ⊆ (𝐴 × 𝐵) → (𝑅 ⊆ 𝑆 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1297   = wceq 1299   ∈ wcel 1448  ∀wral 2375  ∃wrex 2376   ⊆ wss 3021  ⟨cop 3477   × cxp 4475

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-opab 3930  df-xp 4483

This theorem is referenced by: (None)