Theorem eqrelf 37029

Description: The equality connective between relations. (Contributed by Peter Mazsa, 25-Jun-2019.)

Hypotheses

Ref	Expression
eqrelf.1	⊢ Ⅎ𝑥𝐴
eqrelf.2	⊢ Ⅎ𝑥𝐵
eqrelf.3	⊢ Ⅎ𝑦𝐴
eqrelf.4	⊢ Ⅎ𝑦𝐵

Assertion

Ref	Expression
eqrelf	⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem eqrelf

Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqrel 5779	. 2 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑢∀𝑣(⟨𝑢, 𝑣⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐵)))
2		nfv 1918	. . 3 ⊢ Ⅎ𝑢(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
3		nfv 1918	. . 3 ⊢ Ⅎ𝑣(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
4		eqrelf.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
5	4	nfel2 2922	. . . 4 ⊢ Ⅎ𝑥⟨𝑢, 𝑣⟩ ∈ 𝐴
6		eqrelf.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
7	6	nfel2 2922	. . . 4 ⊢ Ⅎ𝑥⟨𝑢, 𝑣⟩ ∈ 𝐵
8	5, 7	nfbi 1907	. . 3 ⊢ Ⅎ𝑥(⟨𝑢, 𝑣⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐵)
9		eqrelf.3	. . . . 5 ⊢ Ⅎ𝑦𝐴
10	9	nfel2 2922	. . . 4 ⊢ Ⅎ𝑦⟨𝑢, 𝑣⟩ ∈ 𝐴
11		eqrelf.4	. . . . 5 ⊢ Ⅎ𝑦𝐵
12	11	nfel2 2922	. . . 4 ⊢ Ⅎ𝑦⟨𝑢, 𝑣⟩ ∈ 𝐵
13	10, 12	nfbi 1907	. . 3 ⊢ Ⅎ𝑦(⟨𝑢, 𝑣⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐵)
14		opeq12 4871	. . . . 5 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩)
15	14	eleq1d 2819	. . . 4 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐴))
16	14	eleq1d 2819	. . . 4 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐵))
17	15, 16	bibi12d 346	. . 3 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ((⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ (⟨𝑢, 𝑣⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐵)))
18	2, 3, 8, 13, 17	cbval2v 2340	. 2 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∀𝑢∀𝑣(⟨𝑢, 𝑣⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐵))
19	1, 18	bitr4di 289	1 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397  ∀wal 1540   = wceq 1542   ∈ wcel 2107  Ⅎwnfc 2884  ⟨cop 4630  Rel wrel 5677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-opab 5207  df-xp 5678  df-rel 5679

This theorem is referenced by:  vvdifopab  37034