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Theorem ssrel 4595
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 3059 . . 3 (𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
21alrimivv 1829 . 2 (𝐴𝐵 → ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
3 eleq1 2178 . . . . . . . . . . 11 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
4 eleq1 2178 . . . . . . . . . . 11 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝐵 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
53, 4imbi12d 233 . . . . . . . . . 10 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝑧𝐴𝑧𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
65biimprcd 159 . . . . . . . . 9 ((⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝐴𝑧𝐵)))
762alimi 1415 . . . . . . . 8 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → ∀𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝐴𝑧𝐵)))
8 19.23vv 1838 . . . . . . . 8 (∀𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝐴𝑧𝐵)) ↔ (∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝐴𝑧𝐵)))
97, 8sylib 121 . . . . . . 7 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝐴𝑧𝐵)))
109com23 78 . . . . . 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 1833 . . . 4 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (∀𝑧(𝑧𝐴 → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩) → ∀𝑧(𝑧𝐴𝑧𝐵)))
13 df-rel 4514 . . . . 5 (Rel 𝐴𝐴 ⊆ (V × V))
14 dfss2 3054 . . . . 5 (𝐴 ⊆ (V × V) ↔ ∀𝑧(𝑧𝐴𝑧 ∈ (V × V)))
15 elvv 4569 . . . . . . 7 (𝑧 ∈ (V × V) ↔ ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩)
1615imbi2i 225 . . . . . 6 ((𝑧𝐴𝑧 ∈ (V × V)) ↔ (𝑧𝐴 → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩))
1716albii 1429 . . . . 5 (∀𝑧(𝑧𝐴𝑧 ∈ (V × V)) ↔ ∀𝑧(𝑧𝐴 → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩))
1813, 14, 173bitri 205 . . . 4 (Rel 𝐴 ↔ ∀𝑧(𝑧𝐴 → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩))
19 dfss2 3054 . . . 4 (𝐴𝐵 ↔ ∀𝑧(𝑧𝐴𝑧𝐵))
2012, 18, 193imtr4g 204 . . 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 142 1 (Rel 𝐴 → (𝐴𝐵 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1312   = wceq 1314  wex 1451  wcel 1463  Vcvv 2658  wss 3039  cop 3498   × cxp 4505  Rel wrel 4512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-opab 3958  df-xp 4513  df-rel 4514
This theorem is referenced by:  eqrel  4596  relssi  4598  relssdv  4599  cotr  4888  cnvsym  4890  intasym  4891  intirr  4893  codir  4895  qfto  4896
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