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Theorem ssrelf 30382
Description: A subclass relationship depends only on a relation's ordered pairs. Theorem 3.2(i) of [Monk1] p. 33. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Thierry Arnoux, 6-Nov-2017.)
Hypotheses
Ref Expression
eqrelrd2.1 𝑥𝜑
eqrelrd2.2 𝑦𝜑
eqrelrd2.3 𝑥𝐴
eqrelrd2.4 𝑦𝐴
eqrelrd2.5 𝑥𝐵
eqrelrd2.6 𝑦𝐵
Assertion
Ref Expression
ssrelf (Rel 𝐴 → (𝐴𝐵 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem ssrelf
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eqrelrd2.3 . . . 4 𝑥𝐴
2 eqrelrd2.5 . . . 4 𝑥𝐵
31, 2nfss 3910 . . 3 𝑥 𝐴𝐵
4 eqrelrd2.4 . . . . 5 𝑦𝐴
5 eqrelrd2.6 . . . . 5 𝑦𝐵
64, 5nfss 3910 . . . 4 𝑦 𝐴𝐵
7 ssel 3911 . . . 4 (𝐴𝐵 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
86, 7alrimi 2212 . . 3 (𝐴𝐵 → ∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
93, 8alrimi 2212 . 2 (𝐴𝐵 → ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))
10 eleq1 2880 . . . . . . . . . . 11 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
11 eleq1 2880 . . . . . . . . . . 11 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝐵 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
1210, 11imbi12d 348 . . . . . . . . . 10 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝑧𝐴𝑧𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
1312biimprcd 253 . . . . . . . . 9 ((⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝐴𝑧𝐵)))
14132alimi 1814 . . . . . . . 8 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → ∀𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝐴𝑧𝐵)))
154nfcri 2946 . . . . . . . . . . . 12 𝑦 𝑧𝐴
165nfcri 2946 . . . . . . . . . . . 12 𝑦 𝑧𝐵
1715, 16nfim 1897 . . . . . . . . . . 11 𝑦(𝑧𝐴𝑧𝐵)
181719.23 2210 . . . . . . . . . 10 (∀𝑦(𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝐴𝑧𝐵)) ↔ (∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝐴𝑧𝐵)))
1918albii 1821 . . . . . . . . 9 (∀𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝐴𝑧𝐵)) ↔ ∀𝑥(∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝐴𝑧𝐵)))
201nfcri 2946 . . . . . . . . . . 11 𝑥 𝑧𝐴
212nfcri 2946 . . . . . . . . . . 11 𝑥 𝑧𝐵
2220, 21nfim 1897 . . . . . . . . . 10 𝑥(𝑧𝐴𝑧𝐵)
232219.23 2210 . . . . . . . . 9 (∀𝑥(∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝐴𝑧𝐵)) ↔ (∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝐴𝑧𝐵)))
2419, 23bitri 278 . . . . . . . 8 (∀𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝐴𝑧𝐵)) ↔ (∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝐴𝑧𝐵)))
2514, 24sylib 221 . . . . . . 7 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧𝐴𝑧𝐵)))
2625com23 86 . . . . . 6 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (𝑧𝐴 → (∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧𝐵)))
2726a2d 29 . . . . 5 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → ((𝑧𝐴 → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩) → (𝑧𝐴𝑧𝐵)))
2827alimdv 1917 . . . 4 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (∀𝑧(𝑧𝐴 → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩) → ∀𝑧(𝑧𝐴𝑧𝐵)))
29 df-rel 5530 . . . . 5 (Rel 𝐴𝐴 ⊆ (V × V))
30 dfss2 3904 . . . . 5 (𝐴 ⊆ (V × V) ↔ ∀𝑧(𝑧𝐴𝑧 ∈ (V × V)))
31 elvv 5594 . . . . . . 7 (𝑧 ∈ (V × V) ↔ ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩)
3231imbi2i 339 . . . . . 6 ((𝑧𝐴𝑧 ∈ (V × V)) ↔ (𝑧𝐴 → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩))
3332albii 1821 . . . . 5 (∀𝑧(𝑧𝐴𝑧 ∈ (V × V)) ↔ ∀𝑧(𝑧𝐴 → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩))
3429, 30, 333bitri 300 . . . 4 (Rel 𝐴 ↔ ∀𝑧(𝑧𝐴 → ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩))
35 dfss2 3904 . . . 4 (𝐴𝐵 ↔ ∀𝑧(𝑧𝐴𝑧𝐵))
3628, 34, 353imtr4g 299 . . 3 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → (Rel 𝐴𝐴𝐵))
3736com12 32 . 2 (Rel 𝐴 → (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) → 𝐴𝐵))
389, 37impbid2 229 1 (Rel 𝐴 → (𝐴𝐵 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1536   = wceq 1538  wex 1781  wnf 1785  wcel 2112  wnfc 2939  Vcvv 3444  wss 3884  cop 4534   × cxp 5521  Rel wrel 5528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-opab 5096  df-xp 5529  df-rel 5530
This theorem is referenced by:  eqrelrd2  30383
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