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Theorem ssrel2 5367
Description: A subclass relationship depends only on a relation's ordered pairs. This version of ssrel 5364 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 3738 . . . 4 (𝑅𝑆 → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆))
21a1d 25 . . 3 (𝑅𝑆 → ((𝑥𝐴𝑦𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆)))
32ralrimivv 3108 . 2 (𝑅𝑆 → ∀𝑥𝐴𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆))
4 eleq1 2827 . . . . . . . . . . 11 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
5 eleq1 2827 . . . . . . . . . . 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 3091 . . . . . . . 8 (∀𝑥𝐴𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → ∀𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)))
9 r19.23v 3161 . . . . . . . . . 10 (∀𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)) ↔ (∃𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)))
109ralbii 3118 . . . . . . . . 9 (∀𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)) ↔ ∀𝑥𝐴 (∃𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)))
11 r19.23v 3161 . . . . . . . . 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 1994 . . . 4 (∀𝑥𝐴𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → (∀𝑧(𝑧𝑅 → ∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩) → ∀𝑧(𝑧𝑅𝑧𝑆)))
17 dfss2 3732 . . . . 5 (𝑅 ⊆ (𝐴 × 𝐵) ↔ ∀𝑧(𝑧𝑅𝑧 ∈ (𝐴 × 𝐵)))
18 elxp2 5289 . . . . . . 7 (𝑧 ∈ (𝐴 × 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩)
1918imbi2i 325 . . . . . 6 ((𝑧𝑅𝑧 ∈ (𝐴 × 𝐵)) ↔ (𝑧𝑅 → ∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩))
2019albii 1896 . . . . 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 3732 . . . 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 1630   = wceq 1632  wcel 2139  wral 3050  wrex 3051  wss 3715  cop 4327   × cxp 5264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-opab 4865  df-xp 5272
This theorem is referenced by:  metuel2  22571  isarchi  30045
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