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	⊢ Ⅎ𝑥𝜑
eqrelrd2.2	⊢ Ⅎ𝑦𝜑
eqrelrd2.3	⊢ Ⅎ𝑥𝐴
eqrelrd2.4	⊢ Ⅎ𝑦𝐴
eqrelrd2.5	⊢ Ⅎ𝑥𝐵
eqrelrd2.6	⊢ Ⅎ𝑦𝐵
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

Step	Hyp	Ref	Expression
1		eqrelrd2.1	. . . 4 ⊢ Ⅎ𝑥𝜑
2		eqrelrd2.2	. . . . 5 ⊢ Ⅎ𝑦𝜑
3		eqrelrd2.7	. . . . 5 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
4	2, 3	alrimi 2120	. . . 4 ⊢ (𝜑 → ∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
5	1, 4	alrimi 2120	. . 3 ⊢ (𝜑 → ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
6	5	adantl 481	. 2 ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
7		eqrelrd2.3	. . . . . 6 ⊢ Ⅎ𝑥𝐴
8		eqrelrd2.4	. . . . . 6 ⊢ Ⅎ𝑦𝐴
9		eqrelrd2.5	. . . . . 6 ⊢ Ⅎ𝑥𝐵
10		eqrelrd2.6	. . . . . 6 ⊢ Ⅎ𝑦𝐵
11	1, 2, 7, 8, 9, 10	ssrelf 29553	. . . . 5 ⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
12	1, 2, 9, 10, 7, 8	ssrelf 29553	. . . . 5 ⊢ (Rel 𝐵 → (𝐵 ⊆ 𝐴 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐵 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
13	11, 12	bi2anan9 935	. . . 4 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴) ↔ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) ∧ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐵 → ⟨𝑥, 𝑦⟩ ∈ 𝐴))))
14		eqss 3651	. . . 4 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
15		2albiim 1857	. . . 4 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) ∧ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐵 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
16	13, 14, 15	3bitr4g 303	. . 3 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
17	16	adantr 480	. 2 ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
18	6, 17	mpbird 247	1 ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383  ∀wal 1521   = wceq 1523  Ⅎwnf 1748   ∈ wcel 2030  Ⅎwnfc 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