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Theorem ssrel 5241
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 4814, ax-nul 4822, ax-pr 4936. (Revised by KP, 25-Oct-2021.)
Assertion
Ref Expression
ssrel (Rel 𝐴 → (𝐴𝐵 ↔ ∀𝑥𝑦(⟨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.
StepHypRef Expression
1 ssel 3630 . . 3 (𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
21alrimivv 1896 . 2 (𝐴𝐵 → ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
3 df-rel 5150 . . . . . . 7 (Rel 𝐴𝐴 ⊆ (V × V))
4 dfss2 3624 . . . . . . 7 (𝐴 ⊆ (V × V) ↔ ∀𝑧(𝑧𝐴𝑧 ∈ (V × V)))
53, 4sylbb 209 . . . . . 6 (Rel 𝐴 → ∀𝑧(𝑧𝐴𝑧 ∈ (V × V)))
6 df-xp 5149 . . . . . . . . . 10 (V × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
7 df-opab 4746 . . . . . . . . . 10 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))}
86, 7eqtri 2673 . . . . . . . . 9 (V × V) = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))}
98abeq2i 2764 . . . . . . . 8 (𝑧 ∈ (V × V) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
10 simpl 472 . . . . . . . . 9 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → 𝑧 = ⟨𝑥, 𝑦⟩)
11102eximi 1803 . . . . . . . 8 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩)
129, 11sylbi 207 . . . . . . 7 (𝑧 ∈ (V × V) → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩)
1312imim2i 16 . . . . . 6 ((𝑧𝐴𝑧 ∈ (V × V)) → (𝑧𝐴 → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩))
145, 13sylg 1790 . . . . 5 (Rel 𝐴 → ∀𝑧(𝑧𝐴 → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩))
15 eleq1 2718 . . . . . . . . . . . 12 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
16 eleq1 2718 . . . . . . . . . . . 12 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝐵 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
1715, 16imbi12d 333 . . . . . . . . . . 11 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝑧𝐴𝑧𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
1817biimprcd 240 . . . . . . . . . 10 ((⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝐴𝑧𝐵)))
19182alimi 1780 . . . . . . . . 9 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → ∀𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝐴𝑧𝐵)))
20 19.23vv 1912 . . . . . . . . 9 (∀𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝐴𝑧𝐵)) ↔ (∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝐴𝑧𝐵)))
2119, 20sylib 208 . . . . . . . 8 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝐴𝑧𝐵)))
2221com23 86 . . . . . . 7 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (𝑧𝐴 → (∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧𝐵)))
2322a2d 29 . . . . . 6 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → ((𝑧𝐴 → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩) → (𝑧𝐴𝑧𝐵)))
2423alimdv 1885 . . . . 5 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (∀𝑧(𝑧𝐴 → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩) → ∀𝑧(𝑧𝐴𝑧𝐵)))
2514, 24syl5 34 . . . 4 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (Rel 𝐴 → ∀𝑧(𝑧𝐴𝑧𝐵)))
26 dfss2 3624 . . . 4 (𝐴𝐵 ↔ ∀𝑧(𝑧𝐴𝑧𝐵))
2725, 26syl6ibr 242 . . 3 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (Rel 𝐴𝐴𝐵))
2827com12 32 . 2 (Rel 𝐴 → (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → 𝐴𝐵))
292, 28impbid2 216 1 (Rel 𝐴 → (𝐴𝐵 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1521   = wceq 1523  wex 1744  wcel 2030  {cab 2637  Vcvv 3231  wss 3607  cop 4216  {copab 4745   × cxp 5141  Rel wrel 5148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-in 3614  df-ss 3621  df-opab 4746  df-xp 5149  df-rel 5150
This theorem is referenced by:  eqrel  5243  relssi  5245  relssdv  5246  cotrg  5542  cnvsym  5545  intasym  5546  intirr  5549  codir  5551  qfto  5552  dfso2  31770  dfpo2  31771  dffun10  32146  imagesset  32185  ssrel3  34208  undmrnresiss  38227  cnvssco  38229
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