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Theorem ssrel2 4710
Description: A subclass relationship depends only on a relation's ordered pairs. This version of ssrel 4708 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 3147 . . . 4 (𝑅𝑆 → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆))
21a1d 22 . . 3 (𝑅𝑆 → ((𝑥𝐴𝑦𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆)))
32ralrimivv 2556 . 2 (𝑅𝑆 → ∀𝑥𝐴𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆))
4 eleq1 2238 . . . . . . . . . . . 12 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
5 eleq1 2238 . . . . . . . . . . . 12 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑆 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑆))
64, 5imbi12d 234 . . . . . . . . . . 11 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝑧𝑅𝑧𝑆) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆)))
76biimprcd 160 . . . . . . . . . 10 ((⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)))
87ralimi 2538 . . . . . . . . 9 (∀𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → ∀𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)))
98ralimi 2538 . . . . . . . 8 (∀𝑥𝐴𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → ∀𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)))
10 r19.23v 2584 . . . . . . . . . 10 (∀𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)) ↔ (∃𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)))
1110ralbii 2481 . . . . . . . . 9 (∀𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)) ↔ ∀𝑥𝐴 (∃𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)))
12 r19.23v 2584 . . . . . . . . 9 (∀𝑥𝐴 (∃𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)) ↔ (∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)))
1311, 12bitri 184 . . . . . . . 8 (∀𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)) ↔ (∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)))
149, 13sylib 122 . . . . . . 7 (∀𝑥𝐴𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → (∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)))
1514com23 78 . . . . . 6 (∀𝑥𝐴𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → (𝑧𝑅 → (∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧𝑆)))
1615a2d 26 . . . . 5 (∀𝑥𝐴𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → ((𝑧𝑅 → ∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩) → (𝑧𝑅𝑧𝑆)))
1716alimdv 1877 . . . 4 (∀𝑥𝐴𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → (∀𝑧(𝑧𝑅 → ∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩) → ∀𝑧(𝑧𝑅𝑧𝑆)))
18 dfss2 3142 . . . . 5 (𝑅 ⊆ (𝐴 × 𝐵) ↔ ∀𝑧(𝑧𝑅𝑧 ∈ (𝐴 × 𝐵)))
19 elxp2 4638 . . . . . . 7 (𝑧 ∈ (𝐴 × 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩)
2019imbi2i 226 . . . . . 6 ((𝑧𝑅𝑧 ∈ (𝐴 × 𝐵)) ↔ (𝑧𝑅 → ∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩))
2120albii 1468 . . . . 5 (∀𝑧(𝑧𝑅𝑧 ∈ (𝐴 × 𝐵)) ↔ ∀𝑧(𝑧𝑅 → ∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩))
2218, 21bitri 184 . . . 4 (𝑅 ⊆ (𝐴 × 𝐵) ↔ ∀𝑧(𝑧𝑅 → ∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩))
23 dfss2 3142 . . . 4 (𝑅𝑆 ↔ ∀𝑧(𝑧𝑅𝑧𝑆))
2417, 22, 233imtr4g 205 . . 3 (∀𝑥𝐴𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → (𝑅 ⊆ (𝐴 × 𝐵) → 𝑅𝑆))
2524com12 30 . 2 (𝑅 ⊆ (𝐴 × 𝐵) → (∀𝑥𝐴𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → 𝑅𝑆))
263, 25impbid2 143 1 (𝑅 ⊆ (𝐴 × 𝐵) → (𝑅𝑆 ↔ ∀𝑥𝐴𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wal 1351   = wceq 1353  wcel 2146  wral 2453  wrex 2454  wss 3127  cop 3592   × cxp 4618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-opab 4060  df-xp 4626
This theorem is referenced by: (None)
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