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Theorem ssrel2 4457
Description: A subclass relationship depends only on a relation's ordered pairs. This version of ssrel 4455 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 2966 . . . 4 (𝑅𝑆 → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆))
21a1d 22 . . 3 (𝑅𝑆 → ((𝑥𝐴𝑦𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆)))
32ralrimivv 2417 . 2 (𝑅𝑆 → ∀𝑥𝐴𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆))
4 eleq1 2116 . . . . . . . . . . . 12 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
5 eleq1 2116 . . . . . . . . . . . 12 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑆 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑆))
64, 5imbi12d 227 . . . . . . . . . . 11 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝑧𝑅𝑧𝑆) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆)))
76biimprcd 153 . . . . . . . . . 10 ((⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)))
87ralimi 2401 . . . . . . . . 9 (∀𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → ∀𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)))
98ralimi 2401 . . . . . . . 8 (∀𝑥𝐴𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → ∀𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)))
10 r19.23v 2442 . . . . . . . . . 10 (∀𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)) ↔ (∃𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)))
1110ralbii 2347 . . . . . . . . 9 (∀𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)) ↔ ∀𝑥𝐴 (∃𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)))
12 r19.23v 2442 . . . . . . . . 9 (∀𝑥𝐴 (∃𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)) ↔ (∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)))
1311, 12bitri 177 . . . . . . . 8 (∀𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)) ↔ (∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)))
149, 13sylib 131 . . . . . . 7 (∀𝑥𝐴𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → (∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝑅𝑧𝑆)))
1514com23 76 . . . . . 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 1775 . . . 4 (∀𝑥𝐴𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆) → (∀𝑧(𝑧𝑅 → ∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩) → ∀𝑧(𝑧𝑅𝑧𝑆)))
18 dfss2 2961 . . . . 5 (𝑅 ⊆ (𝐴 × 𝐵) ↔ ∀𝑧(𝑧𝑅𝑧 ∈ (𝐴 × 𝐵)))
19 elxp2 4390 . . . . . . 7 (𝑧 ∈ (𝐴 × 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩)
2019imbi2i 219 . . . . . 6 ((𝑧𝑅𝑧 ∈ (𝐴 × 𝐵)) ↔ (𝑧𝑅 → ∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩))
2120albii 1375 . . . . 5 (∀𝑧(𝑧𝑅𝑧 ∈ (𝐴 × 𝐵)) ↔ ∀𝑧(𝑧𝑅 → ∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩))
2218, 21bitri 177 . . . 4 (𝑅 ⊆ (𝐴 × 𝐵) ↔ ∀𝑧(𝑧𝑅 → ∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩))
23 dfss2 2961 . . . 4 (𝑅𝑆 ↔ ∀𝑧(𝑧𝑅𝑧𝑆))
2417, 22, 233imtr4g 198 . . 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 135 1 (𝑅 ⊆ (𝐴 × 𝐵) → (𝑅𝑆 ↔ ∀𝑥𝐴𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1257   = wceq 1259  wcel 1409  wral 2323  wrex 2324  wss 2944  cop 3405   × cxp 4370
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-opab 3846  df-xp 4378
This theorem is referenced by: (None)
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