Theorem eqrelf 39382

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 5765	. 2 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑢∀𝑣(⟨𝑢, 𝑣⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐵)))
2		nfv 1949	. . 3 ⊢ Ⅎ𝑢(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
3		nfv 1949	. . 3 ⊢ Ⅎ𝑣(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
4		eqrelf.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
5	4	nfel2 2944	. . . 4 ⊢ Ⅎ𝑥⟨𝑢, 𝑣⟩ ∈ 𝐴
6		eqrelf.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
7	6	nfel2 2944	. . . 4 ⊢ Ⅎ𝑥⟨𝑢, 𝑣⟩ ∈ 𝐵
8	5, 7	nfbi 1938	. . 3 ⊢ Ⅎ𝑥(⟨𝑢, 𝑣⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐵)
9		eqrelf.3	. . . . 5 ⊢ Ⅎ𝑦𝐴
10	9	nfel2 2944	. . . 4 ⊢ Ⅎ𝑦⟨𝑢, 𝑣⟩ ∈ 𝐴
11		eqrelf.4	. . . . 5 ⊢ Ⅎ𝑦𝐵
12	11	nfel2 2944	. . . 4 ⊢ Ⅎ𝑦⟨𝑢, 𝑣⟩ ∈ 𝐵
13	10, 12	nfbi 1938	. . 3 ⊢ Ⅎ𝑦(⟨𝑢, 𝑣⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐵)
14		opeq12 4845	. . . . 5 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩)
15	14	eleq1d 2849	. . . 4 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐴))
16	14	eleq1d 2849	. . . 4 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐵))
17	15, 16	bibi12d 348	. . 3 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ((⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ (⟨𝑢, 𝑣⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐵)))
18	2, 3, 8, 13, 17	cbval2v 2376	. 2 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∀𝑢∀𝑣(⟨𝑢, 𝑣⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐵))
19	1, 18	bitr4di 292	1 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2147  Ⅎwnfc 2911  ⟨cop 4838  Rel wrel 5660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-opab 5171  df-xp 5661  df-rel 5662

This theorem is used by:  vvdifopab  39389