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Theorem ssrelrel 4639
 Description: A subclass relationship determined by ordered triples. Use relrelss 5065 to express the antecedent in terms of the relation predicate. (Contributed by NM, 17-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
ssrelrel (𝐴 ⊆ ((V × V) × V) → (𝐴𝐵 ↔ ∀𝑥𝑦𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧

Proof of Theorem ssrelrel
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 ssel 3091 . . . 4 (𝐴𝐵 → (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵))
21alrimiv 1846 . . 3 (𝐴𝐵 → ∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵))
32alrimivv 1847 . 2 (𝐴𝐵 → ∀𝑥𝑦𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵))
4 elvvv 4602 . . . . . . . 8 (𝑤 ∈ ((V × V) × V) ↔ ∃𝑥𝑦𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
5 eleq1 2202 . . . . . . . . . . . . . 14 (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤𝐴 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴))
6 eleq1 2202 . . . . . . . . . . . . . 14 (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤𝐵 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵))
75, 6imbi12d 233 . . . . . . . . . . . . 13 (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → ((𝑤𝐴𝑤𝐵) ↔ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵)))
87biimprcd 159 . . . . . . . . . . . 12 ((⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤𝐴𝑤𝐵)))
98alimi 1431 . . . . . . . . . . 11 (∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) → ∀𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤𝐴𝑤𝐵)))
10 19.23v 1855 . . . . . . . . . . 11 (∀𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤𝐴𝑤𝐵)) ↔ (∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤𝐴𝑤𝐵)))
119, 10sylib 121 . . . . . . . . . 10 (∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) → (∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤𝐴𝑤𝐵)))
12112alimi 1432 . . . . . . . . 9 (∀𝑥𝑦𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) → ∀𝑥𝑦(∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤𝐴𝑤𝐵)))
13 19.23vv 1856 . . . . . . . . 9 (∀𝑥𝑦(∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤𝐴𝑤𝐵)) ↔ (∃𝑥𝑦𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤𝐴𝑤𝐵)))
1412, 13sylib 121 . . . . . . . 8 (∀𝑥𝑦𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) → (∃𝑥𝑦𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤𝐴𝑤𝐵)))
154, 14syl5bi 151 . . . . . . 7 (∀𝑥𝑦𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) → (𝑤 ∈ ((V × V) × V) → (𝑤𝐴𝑤𝐵)))
1615com23 78 . . . . . 6 (∀𝑥𝑦𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) → (𝑤𝐴 → (𝑤 ∈ ((V × V) × V) → 𝑤𝐵)))
1716a2d 26 . . . . 5 (∀𝑥𝑦𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) → ((𝑤𝐴𝑤 ∈ ((V × V) × V)) → (𝑤𝐴𝑤𝐵)))
1817alimdv 1851 . . . 4 (∀𝑥𝑦𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) → (∀𝑤(𝑤𝐴𝑤 ∈ ((V × V) × V)) → ∀𝑤(𝑤𝐴𝑤𝐵)))
19 dfss2 3086 . . . 4 (𝐴 ⊆ ((V × V) × V) ↔ ∀𝑤(𝑤𝐴𝑤 ∈ ((V × V) × V)))
20 dfss2 3086 . . . 4 (𝐴𝐵 ↔ ∀𝑤(𝑤𝐴𝑤𝐵))
2118, 19, 203imtr4g 204 . . 3 (∀𝑥𝑦𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) → (𝐴 ⊆ ((V × V) × V) → 𝐴𝐵))
2221com12 30 . 2 (𝐴 ⊆ ((V × V) × V) → (∀𝑥𝑦𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵) → 𝐴𝐵))
233, 22impbid2 142 1 (𝐴 ⊆ ((V × V) × V) → (𝐴𝐵 ↔ ∀𝑥𝑦𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1329   = wceq 1331  ∃wex 1468   ∈ wcel 1480  Vcvv 2686   ⊆ wss 3071  ⟨cop 3530   × cxp 4537 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-opab 3990  df-xp 4545 This theorem is referenced by:  eqrelrel  4640
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