Theorem ssrel 5745

Description: A subclass relationship depends only on a relation's ordered pairs. Theorem 3.2(i) of [Monk1] p. 33. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Remove dependency on ax-sep 5251, ax-nul 5261, ax-pr 5387. (Revised by KP, 25-Oct-2021.) Remove dependency on ax-12 2178. (Revised by SN, 11-Dec-2024.)

Assertion

Ref	Expression
ssrel	⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))

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

Proof of Theorem ssrel

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3940	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
2	1	alrimivv 1928	. 2 ⊢ (𝐴 ⊆ 𝐵 → ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
3		df-rel 5645	. . . . . . 7 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
4		df-ss 3931	. . . . . . 7 ⊢ (𝐴 ⊆ (V × V) ↔ ∀𝑧(𝑧 ∈ 𝐴 → 𝑧 ∈ (V × V)))
5	3, 4	sylbb 219	. . . . . 6 ⊢ (Rel 𝐴 → ∀𝑧(𝑧 ∈ 𝐴 → 𝑧 ∈ (V × V)))
6		elopabw 5486	. . . . . . . . . 10 ⊢ (𝑧 ∈ V → (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))))
7	6	elv 3452	. . . . . . . . 9 ⊢ (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
8		simpl 482	. . . . . . . . . 10 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → 𝑧 = ⟨𝑥, 𝑦⟩)
9	8	2eximi 1836	. . . . . . . . 9 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩)
10	7, 9	sylbi 217	. . . . . . . 8 ⊢ (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩)
11		df-xp 5644	. . . . . . . 8 ⊢ (V × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
12	10, 11	eleq2s 2846	. . . . . . 7 ⊢ (𝑧 ∈ (V × V) → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩)
13	12	imim2i 16	. . . . . 6 ⊢ ((𝑧 ∈ 𝐴 → 𝑧 ∈ (V × V)) → (𝑧 ∈ 𝐴 → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩))
14	5, 13	sylg 1823	. . . . 5 ⊢ (Rel 𝐴 → ∀𝑧(𝑧 ∈ 𝐴 → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩))
15		eleq1 2816	. . . . . . . . . . . 12 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
16		eleq1 2816	. . . . . . . . . . . 12 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝐵 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
17	15, 16	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
18	17	biimprcd 250	. . . . . . . . . 10 ⊢ ((⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵)))
19	18	2alimi 1812	. . . . . . . . 9 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → ∀𝑥∀𝑦(𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵)))
20		19.23vv 1943	. . . . . . . . 9 ⊢ (∀𝑥∀𝑦(𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵)) ↔ (∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵)))
21	19, 20	sylib 218	. . . . . . . 8 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵)))
22	21	com23 86	. . . . . . 7 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (𝑧 ∈ 𝐴 → (∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 ∈ 𝐵)))
23	22	a2d 29	. . . . . 6 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → ((𝑧 ∈ 𝐴 → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩) → (𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵)))
24	23	alimdv 1916	. . . . 5 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (∀𝑧(𝑧 ∈ 𝐴 → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩) → ∀𝑧(𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵)))
25	14, 24	syl5 34	. . . 4 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (Rel 𝐴 → ∀𝑧(𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵)))
26		df-ss 3931	. . . 4 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑧(𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵))
27	25, 26	imbitrrdi 252	. . 3 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (Rel 𝐴 → 𝐴 ⊆ 𝐵))
28	27	com12 32	. 2 ⊢ (Rel 𝐴 → (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → 𝐴 ⊆ 𝐵))
29	2, 28	impbid2 226	1 ⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3447   ⊆ wss 3914  ⟨cop 4595  {copab 5169   × cxp 5636  Rel wrel 5643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3449  df-ss 3931  df-opab 5170  df-xp 5644  df-rel 5645

This theorem is referenced by:  eqrel  5747  ssrel3  5749  relssi  5750  relssdv  5751  cotrgOLDOLD  6082  cnvsymOLDOLD  6087  intasym  6088  intirr  6091  codir  6093  qfto  6094  dfpo2  6269  ssttrcl  9668  ttrclss  9673  dfso2  35742  dffun10  35902  imagesset  35941  undmrnresiss  43593  cnvssco  43595  joindm2  48956  meetdm2  48958