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Theorem eqrelrd2 30295
Description: A version of eqrelrdv2 5661 with explicit non-free declarations. (Contributed by Thierry Arnoux, 28-Aug-2017.)
Hypotheses
Ref Expression
eqrelrd2.1 𝑥𝜑
eqrelrd2.2 𝑦𝜑
eqrelrd2.3 𝑥𝐴
eqrelrd2.4 𝑦𝐴
eqrelrd2.5 𝑥𝐵
eqrelrd2.6 𝑦𝐵
eqrelrd2.7 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
Assertion
Ref Expression
eqrelrd2 (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem eqrelrd2
StepHypRef Expression
1 eqrelrd2.1 . . . 4 𝑥𝜑
2 eqrelrd2.2 . . . . 5 𝑦𝜑
3 eqrelrd2.7 . . . . 5 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
42, 3alrimi 2203 . . . 4 (𝜑 → ∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
51, 4alrimi 2203 . . 3 (𝜑 → ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
65adantl 482 . 2 (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
7 eqrelrd2.3 . . . . . 6 𝑥𝐴
8 eqrelrd2.4 . . . . . 6 𝑦𝐴
9 eqrelrd2.5 . . . . . 6 𝑥𝐵
10 eqrelrd2.6 . . . . . 6 𝑦𝐵
111, 2, 7, 8, 9, 10ssrelf 30294 . . . . 5 (Rel 𝐴 → (𝐴𝐵 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
121, 2, 9, 10, 7, 8ssrelf 30294 . . . . 5 (Rel 𝐵 → (𝐵𝐴 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐵 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
1311, 12bi2anan9 635 . . . 4 ((Rel 𝐴 ∧ Rel 𝐵) → ((𝐴𝐵𝐵𝐴) ↔ (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) ∧ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐵 → ⟨𝑥, 𝑦⟩ ∈ 𝐴))))
14 eqss 3979 . . . 4 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
15 2albiim 1882 . . . 4 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) ∧ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐵 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
1613, 14, 153bitr4g 315 . . 3 ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
1716adantr 481 . 2 (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → (𝐴 = 𝐵 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
186, 17mpbird 258 1 (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1526   = wceq 1528  wnf 1775  wcel 2105  wnfc 2958  wss 3933  cop 4563  Rel wrel 5553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-opab 5120  df-xp 5554  df-rel 5555
This theorem is referenced by:  fpwrelmap  30395
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