Theorem ssrel 5769

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 5256, ax-nul 5268, ax-pr 5404. (Revised by KP, 25-Oct-2021.) Remove dependency on ax-12 2211. (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 3930	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
2	1	alrimivv 1956	. 2 ⊢ (𝐴 ⊆ 𝐵 → ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
3		df-rel 5668	. . . . . . 7 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
4		df-ss 3921	. . . . . . 7 ⊢ (𝐴 ⊆ (V × V) ↔ ∀𝑧(𝑧 ∈ 𝐴 → 𝑧 ∈ (V × V)))
5	3, 4	sylbb 222	. . . . . 6 ⊢ (Rel 𝐴 → ∀𝑧(𝑧 ∈ 𝐴 → 𝑧 ∈ (V × V)))
6		elopabw 5510	. . . . . . . . . 10 ⊢ (𝑧 ∈ V → (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))))
7	6	elv 3458	. . . . . . . . 9 ⊢ (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
8		simpl 487	. . . . . . . . . 10 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → 𝑧 = ⟨𝑥, 𝑦⟩)
9	8	2eximi 1864	. . . . . . . . 9 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩)
10	7, 9	sylbi 220	. . . . . . . 8 ⊢ (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩)
11		df-xp 5667	. . . . . . . 8 ⊢ (V × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
12	10, 11	eleq2s 2879	. . . . . . 7 ⊢ (𝑧 ∈ (V × V) → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩)
13	12	imim2i 17	. . . . . 6 ⊢ ((𝑧 ∈ 𝐴 → 𝑧 ∈ (V × V)) → (𝑧 ∈ 𝐴 → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩))
14	5, 13	sylg 1851	. . . . 5 ⊢ (Rel 𝐴 → ∀𝑧(𝑧 ∈ 𝐴 → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩))
15		eleq1 2849	. . . . . . . . . . . 12 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
16		eleq1 2849	. . . . . . . . . . . 12 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝐵 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
17	15, 16	imbi12d 347	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
18	17	biimprcd 253	. . . . . . . . . 10 ⊢ ((⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵)))
19	18	2alimi 1840	. . . . . . . . 9 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → ∀𝑥∀𝑦(𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵)))
20		19.23vv 1971	. . . . . . . . 9 ⊢ (∀𝑥∀𝑦(𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵)) ↔ (∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵)))
21	19, 20	sylib 221	. . . . . . . 8 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵)))
22	21	com23 87	. . . . . . 7 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (𝑧 ∈ 𝐴 → (∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 ∈ 𝐵)))
23	22	a2d 30	. . . . . 6 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → ((𝑧 ∈ 𝐴 → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩) → (𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵)))
24	23	alimdv 1944	. . . . 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 35	. . . 4 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (Rel 𝐴 → ∀𝑧(𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵)))
26		df-ss 3921	. . . 4 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑧(𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵))
27	25, 26	imbitrrdi 255	. . 3 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (Rel 𝐴 → 𝐴 ⊆ 𝐵))
28	27	com12 33	. 2 ⊢ (Rel 𝐴 → (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → 𝐴 ⊆ 𝐵))
29	2, 28	impbid2 229	1 ⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1566   = wceq 1568  ∃wex 1807   ∈ wcel 2141  Vcvv 3453   ⊆ wss 3904  ⟨cop 4594  {copab 5172   × cxp 5659  Rel wrel 5666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1571  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-ss 3921  df-opab 5173  df-xp 5667  df-rel 5668

This theorem is referenced by:  eqrel  5770  ssrel3  5772  relssi  5773  relssdv  5774  intasym  6115  intirr  6118  codir  6120  qfto  6121  dfpo2  6297  ssttrcl  9683  ttrclss  9688  dfso2  36213  dffun10  36370  imagesset  36411  undmrnresiss  44300  cnvssco  44302  joindm2  49713  meetdm2  49715