Theorem ssrel 5692

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 5227, ax-nul 5234, ax-pr 5356. (Revised by KP, 25-Oct-2021.)

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 3919	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
2	1	alrimivv 1935	. 2 ⊢ (𝐴 ⊆ 𝐵 → ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
3		df-rel 5596	. . . . . . 7 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
4		dfss2 3912	. . . . . . 7 ⊢ (𝐴 ⊆ (V × V) ↔ ∀𝑧(𝑧 ∈ 𝐴 → 𝑧 ∈ (V × V)))
5	3, 4	sylbb 218	. . . . . 6 ⊢ (Rel 𝐴 → ∀𝑧(𝑧 ∈ 𝐴 → 𝑧 ∈ (V × V)))
6		df-xp 5595	. . . . . . . . . 10 ⊢ (V × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
7		df-opab 5142	. . . . . . . . . 10 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))}
8	6, 7	eqtri 2768	. . . . . . . . 9 ⊢ (V × V) = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))}
9	8	abeq2i 2877	. . . . . . . 8 ⊢ (𝑧 ∈ (V × V) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
10		simpl 483	. . . . . . . . 9 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → 𝑧 = ⟨𝑥, 𝑦⟩)
11	10	2eximi 1842	. . . . . . . 8 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩)
12	9, 11	sylbi 216	. . . . . . 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 1829	. . . . 5 ⊢ (Rel 𝐴 → ∀𝑧(𝑧 ∈ 𝐴 → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩))
15		eleq1 2828	. . . . . . . . . . . 12 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
16		eleq1 2828	. . . . . . . . . . . 12 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝐵 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
17	15, 16	imbi12d 345	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
18	17	biimprcd 249	. . . . . . . . . 10 ⊢ ((⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵)))
19	18	2alimi 1819	. . . . . . . . 9 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → ∀𝑥∀𝑦(𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵)))
20		19.23vv 1950	. . . . . . . . 9 ⊢ (∀𝑥∀𝑦(𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵)) ↔ (∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵)))
21	19, 20	sylib 217	. . . . . . . 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 1923	. . . . 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		dfss2 3912	. . . 4 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑧(𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵))
27	25, 26	syl6ibr 251	. . 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 225	1 ⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1540   = wceq 1542  ∃wex 1786   ∈ wcel 2110  {cab 2717  Vcvv 3431   ⊆ wss 3892  ⟨cop 4573  {copab 5141   × cxp 5587  Rel wrel 5594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-12 2175  ax-ext 2711

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1545  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-v 3433  df-in 3899  df-ss 3909  df-opab 5142  df-xp 5595  df-rel 5596

This theorem is referenced by:  eqrel  5693  relssi  5695  relssdv  5696  cotrg  6014  cnvsym  6017  intasym  6018  intirr  6021  codir  6023  qfto  6024  dfpo2  6197  ssttrcl  9443  ttrclss  9448  dfso2  33710  dffun10  34204  imagesset  34243  ssrel3  36422  undmrnresiss  41174  cnvssco  41176  joindm2  46223  meetdm2  46225