Theorem ssrel2 4713

Description: A subclass relationship depends only on a relation's ordered pairs. This version of ssrel 4711 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 3149	. . . 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 2558	. 2 ⊢ (𝑅 ⊆ 𝑆 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆))
4		eleq1 2240	. . . . . . . . . . . 12 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
5		eleq1 2240	. . . . . . . . . . . 12 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑆 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑆))
6	4, 5	imbi12d 234	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆)))
7	6	biimprcd 160	. . . . . . . . . 10 ⊢ ((⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆)))
8	7	ralimi 2540	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → ∀𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆)))
9	8	ralimi 2540	. . . . . . . 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 2586	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆)) ↔ (∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆)))
11	10	ralbii 2483	. . . . . . . . 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 2586	. . . . . . . . 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 184	. . . . . . . 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 122	. . . . . . 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 1879	. . . 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 3144	. . . . 5 ⊢ (𝑅 ⊆ (𝐴 × 𝐵) ↔ ∀𝑧(𝑧 ∈ 𝑅 → 𝑧 ∈ (𝐴 × 𝐵)))
19		elxp2 4641	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩)
20	19	imbi2i 226	. . . . . 6 ⊢ ((𝑧 ∈ 𝑅 → 𝑧 ∈ (𝐴 × 𝐵)) ↔ (𝑧 ∈ 𝑅 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩))
21	20	albii 1470	. . . . 5 ⊢ (∀𝑧(𝑧 ∈ 𝑅 → 𝑧 ∈ (𝐴 × 𝐵)) ↔ ∀𝑧(𝑧 ∈ 𝑅 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩))
22	18, 21	bitri 184	. . . 4 ⊢ (𝑅 ⊆ (𝐴 × 𝐵) ↔ ∀𝑧(𝑧 ∈ 𝑅 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟨𝑥, 𝑦⟩))
23		dfss2 3144	. . . 4 ⊢ (𝑅 ⊆ 𝑆 ↔ ∀𝑧(𝑧 ∈ 𝑅 → 𝑧 ∈ 𝑆))
24	17, 22, 23	3imtr4g 205	. . 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 143	1 ⊢ (𝑅 ⊆ (𝐴 × 𝐵) → (𝑅 ⊆ 𝑆 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1351   = wceq 1353   ∈ wcel 2148  ∀wral 2455  ∃wrex 2456   ⊆ wss 3129  ⟨cop 3594   × cxp 4621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-opab 4062  df-xp 4629

This theorem is referenced by: (None)