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Theorem ssrel2 5122
Description: A subclass relationship depends only on a relation's ordered pairs. This version of ssrel 5120 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 3561 . . . 4 (𝑅𝑆 → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆))
21a1d 25 . . 3 (𝑅𝑆 → ((𝑥𝐴𝑦𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆)))
32ralrimivv 2952 . 2 (𝑅𝑆 → ∀𝑥𝐴𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆))
4 eleq1 2675 . . . . . . . . . . . 12 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
5 eleq1 2675 . . . . . . . . . . . 12 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑆 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑆))
64, 5imbi12d 332 . . . . . . . . . . 11 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝑧𝑅𝑧𝑆) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆)))
76biimprcd 238 . . . . . . . . . 10 ((⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)))
87ralimi 2935 . . . . . . . . 9 (∀𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → ∀𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)))
98ralimi 2935 . . . . . . . 8 (∀𝑥𝐴𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → ∀𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)))
10 r19.23v 3004 . . . . . . . . . 10 (∀𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)) ↔ (∃𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)))
1110ralbii 2962 . . . . . . . . 9 (∀𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)) ↔ ∀𝑥𝐴 (∃𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)))
12 r19.23v 3004 . . . . . . . . 9 (∀𝑥𝐴 (∃𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)) ↔ (∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)))
1311, 12bitri 262 . . . . . . . 8 (∀𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)) ↔ (∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)))
149, 13sylib 206 . . . . . . 7 (∀𝑥𝐴𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → (∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)))
1514com23 83 . . . . . 6 (∀𝑥𝐴𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → (𝑧𝑅 → (∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧𝑆)))
1615a2d 29 . . . . 5 (∀𝑥𝐴𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → ((𝑧𝑅 → ∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩) → (𝑧𝑅𝑧𝑆)))
1716alimdv 1831 . . . 4 (∀𝑥𝐴𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → (∀𝑧(𝑧𝑅 → ∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩) → ∀𝑧(𝑧𝑅𝑧𝑆)))
18 dfss2 3556 . . . . 5 (𝑅 ⊆ (𝐴 × 𝐵) ↔ ∀𝑧(𝑧𝑅𝑧 ∈ (𝐴 × 𝐵)))
19 elxp2 5046 . . . . . . 7 (𝑧 ∈ (𝐴 × 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩)
2019imbi2i 324 . . . . . 6 ((𝑧𝑅𝑧 ∈ (𝐴 × 𝐵)) ↔ (𝑧𝑅 → ∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩))
2120albii 1736 . . . . 5 (∀𝑧(𝑧𝑅𝑧 ∈ (𝐴 × 𝐵)) ↔ ∀𝑧(𝑧𝑅 → ∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩))
2218, 21bitri 262 . . . 4 (𝑅 ⊆ (𝐴 × 𝐵) ↔ ∀𝑧(𝑧𝑅 → ∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩))
23 dfss2 3556 . . . 4 (𝑅𝑆 ↔ ∀𝑧(𝑧𝑅𝑧𝑆))
2417, 22, 233imtr4g 283 . . 3 (∀𝑥𝐴𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → (𝑅 ⊆ (𝐴 × 𝐵) → 𝑅𝑆))
2524com12 32 . 2 (𝑅 ⊆ (𝐴 × 𝐵) → (∀𝑥𝐴𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → 𝑅𝑆))
263, 25impbid2 214 1 (𝑅 ⊆ (𝐴 × 𝐵) → (𝑅𝑆 ↔ ∀𝑥𝐴𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wal 1472   = wceq 1474  wcel 1976  wral 2895  wrex 2896  wss 3539  cop 4130   × cxp 5026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-opab 4638  df-xp 5034
This theorem is referenced by:  metuel2  22121  isarchi  28873
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