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Theorem ssrel2 5244
Description: A subclass relationship depends only on a relation's ordered pairs. This version of ssrel 5241 is restricted to the relation's domain. (Contributed by Thierry Arnoux, 25-Jan-2018.)
Assertion
Ref Expression
ssrel2 (𝑅 ⊆ (𝐴 × 𝐵) → (𝑅𝑆 ↔ ∀𝑥𝐴𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦

Proof of Theorem ssrel2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 ssel 3630 . . . 4 (𝑅𝑆 → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆))
21a1d 25 . . 3 (𝑅𝑆 → ((𝑥𝐴𝑦𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆)))
32ralrimivv 2999 . 2 (𝑅𝑆 → ∀𝑥𝐴𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆))
4 eleq1 2718 . . . . . . . . . . 11 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
5 eleq1 2718 . . . . . . . . . . 11 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑆 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑆))
64, 5imbi12d 333 . . . . . . . . . 10 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝑧𝑅𝑧𝑆) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆)))
76biimprcd 240 . . . . . . . . 9 ((⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)))
872ralimi 2982 . . . . . . . 8 (∀𝑥𝐴𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → ∀𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)))
9 r19.23v 3052 . . . . . . . . . 10 (∀𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)) ↔ (∃𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)))
109ralbii 3009 . . . . . . . . 9 (∀𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)) ↔ ∀𝑥𝐴 (∃𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)))
11 r19.23v 3052 . . . . . . . . 9 (∀𝑥𝐴 (∃𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)) ↔ (∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)))
1210, 11bitri 264 . . . . . . . 8 (∀𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)) ↔ (∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)))
138, 12sylib 208 . . . . . . 7 (∀𝑥𝐴𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → (∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)))
1413com23 86 . . . . . 6 (∀𝑥𝐴𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → (𝑧𝑅 → (∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧𝑆)))
1514a2d 29 . . . . 5 (∀𝑥𝐴𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → ((𝑧𝑅 → ∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩) → (𝑧𝑅𝑧𝑆)))
1615alimdv 1885 . . . 4 (∀𝑥𝐴𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → (∀𝑧(𝑧𝑅 → ∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩) → ∀𝑧(𝑧𝑅𝑧𝑆)))
17 dfss2 3624 . . . . 5 (𝑅 ⊆ (𝐴 × 𝐵) ↔ ∀𝑧(𝑧𝑅𝑧 ∈ (𝐴 × 𝐵)))
18 elxp2 5166 . . . . . . 7 (𝑧 ∈ (𝐴 × 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩)
1918imbi2i 325 . . . . . 6 ((𝑧𝑅𝑧 ∈ (𝐴 × 𝐵)) ↔ (𝑧𝑅 → ∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩))
2019albii 1787 . . . . 5 (∀𝑧(𝑧𝑅𝑧 ∈ (𝐴 × 𝐵)) ↔ ∀𝑧(𝑧𝑅 → ∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩))
2117, 20bitri 264 . . . 4 (𝑅 ⊆ (𝐴 × 𝐵) ↔ ∀𝑧(𝑧𝑅 → ∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩))
22 dfss2 3624 . . . 4 (𝑅𝑆 ↔ ∀𝑧(𝑧𝑅𝑧𝑆))
2316, 21, 223imtr4g 285 . . 3 (∀𝑥𝐴𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → (𝑅 ⊆ (𝐴 × 𝐵) → 𝑅𝑆))
2423com12 32 . 2 (𝑅 ⊆ (𝐴 × 𝐵) → (∀𝑥𝐴𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → 𝑅𝑆))
253, 24impbid2 216 1 (𝑅 ⊆ (𝐴 × 𝐵) → (𝑅𝑆 ↔ ∀𝑥𝐴𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1521   = wceq 1523  wcel 2030  wral 2941  wrex 2942  wss 3607  cop 4216   × cxp 5141
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  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-opab 4746  df-xp 5149
This theorem is referenced by:  metuel2  22417  isarchi  29864
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