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Theorem ssrel 4514
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.)
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 3017 . . 3 (𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
21alrimivv 1803 . 2 (𝐴𝐵 → ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
3 eleq1 2150 . . . . . . . . . . 11 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
4 eleq1 2150 . . . . . . . . . . 11 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝐵 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
53, 4imbi12d 232 . . . . . . . . . 10 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝑧𝐴𝑧𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
65biimprcd 158 . . . . . . . . 9 ((⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝐴𝑧𝐵)))
762alimi 1390 . . . . . . . 8 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → ∀𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝐴𝑧𝐵)))
8 19.23vv 1812 . . . . . . . 8 (∀𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝐴𝑧𝐵)) ↔ (∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝐴𝑧𝐵)))
97, 8sylib 120 . . . . . . 7 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝐴𝑧𝐵)))
109com23 77 . . . . . 6 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (𝑧𝐴 → (∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧𝐵)))
1110a2d 26 . . . . 5 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → ((𝑧𝐴 → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩) → (𝑧𝐴𝑧𝐵)))
1211alimdv 1807 . . . 4 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (∀𝑧(𝑧𝐴 → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩) → ∀𝑧(𝑧𝐴𝑧𝐵)))
13 df-rel 4435 . . . . 5 (Rel 𝐴𝐴 ⊆ (V × V))
14 dfss2 3012 . . . . 5 (𝐴 ⊆ (V × V) ↔ ∀𝑧(𝑧𝐴𝑧 ∈ (V × V)))
15 elvv 4488 . . . . . . 7 (𝑧 ∈ (V × V) ↔ ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩)
1615imbi2i 224 . . . . . 6 ((𝑧𝐴𝑧 ∈ (V × V)) ↔ (𝑧𝐴 → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩))
1716albii 1404 . . . . 5 (∀𝑧(𝑧𝐴𝑧 ∈ (V × V)) ↔ ∀𝑧(𝑧𝐴 → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩))
1813, 14, 173bitri 204 . . . 4 (Rel 𝐴 ↔ ∀𝑧(𝑧𝐴 → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩))
19 dfss2 3012 . . . 4 (𝐴𝐵 ↔ ∀𝑧(𝑧𝐴𝑧𝐵))
2012, 18, 193imtr4g 203 . . 3 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (Rel 𝐴𝐴𝐵))
2120com12 30 . 2 (Rel 𝐴 → (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → 𝐴𝐵))
222, 21impbid2 141 1 (Rel 𝐴 → (𝐴𝐵 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1287   = wceq 1289  wex 1426  wcel 1438  Vcvv 2619  wss 2997  cop 3444   × cxp 4426  Rel wrel 4433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-opab 3892  df-xp 4434  df-rel 4435
This theorem is referenced by:  eqrel  4515  relssi  4517  relssdv  4518  cotr  4800  cnvsym  4802  intasym  4803  intirr  4805  codir  4807  qfto  4808
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