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Theorem eqrelrd2 29554
Description: A version of eqrelrdv2 5253 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 2120 . . . 4 (𝜑 → ∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
51, 4alrimi 2120 . . 3 (𝜑 → ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
65adantl 481 . 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 29553 . . . . 5 (Rel 𝐴 → (𝐴𝐵 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
121, 2, 9, 10, 7, 8ssrelf 29553 . . . . 5 (Rel 𝐵 → (𝐵𝐴 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐵 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
1311, 12bi2anan9 935 . . . 4 ((Rel 𝐴 ∧ Rel 𝐵) → ((𝐴𝐵𝐵𝐴) ↔ (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) ∧ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐵 → ⟨𝑥, 𝑦⟩ ∈ 𝐴))))
14 eqss 3651 . . . 4 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
15 2albiim 1857 . . . 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 303 . . 3 ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
1716adantr 480 . 2 (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → (𝐴 = 𝐵 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
186, 17mpbird 247 1 (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1521   = wceq 1523  wnf 1748  wcel 2030  wnfc 2780  wss 3607  cop 4216  Rel wrel 5148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-opab 4746  df-xp 5149  df-rel 5150
This theorem is referenced by:  fpwrelmap  29636
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