Theorem eqrelf 38939

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 5773	. 2 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑢∀𝑣(⟨𝑢, 𝑣⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐵)))
2		nfv 1947	. . 3 ⊢ Ⅎ𝑢(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
3		nfv 1947	. . 3 ⊢ Ⅎ𝑣(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
4		eqrelf.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
5	4	nfel2 2946	. . . 4 ⊢ Ⅎ𝑥⟨𝑢, 𝑣⟩ ∈ 𝐴
6		eqrelf.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
7	6	nfel2 2946	. . . 4 ⊢ Ⅎ𝑥⟨𝑢, 𝑣⟩ ∈ 𝐵
8	5, 7	nfbi 1936	. . 3 ⊢ Ⅎ𝑥(⟨𝑢, 𝑣⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐵)
9		eqrelf.3	. . . . 5 ⊢ Ⅎ𝑦𝐴
10	9	nfel2 2946	. . . 4 ⊢ Ⅎ𝑦⟨𝑢, 𝑣⟩ ∈ 𝐴
11		eqrelf.4	. . . . 5 ⊢ Ⅎ𝑦𝐵
12	11	nfel2 2946	. . . 4 ⊢ Ⅎ𝑦⟨𝑢, 𝑣⟩ ∈ 𝐵
13	10, 12	nfbi 1936	. . 3 ⊢ Ⅎ𝑦(⟨𝑢, 𝑣⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐵)
14		opeq12 4843	. . . . 5 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩)
15	14	eleq1d 2851	. . . 4 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐴))
16	14	eleq1d 2851	. . . 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 2378	. 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 2146  Ⅎwnfc 2913  ⟨cop 4598  Rel wrel 5669

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738

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 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-ss 3925  df-nul 4290  df-if 4491  df-sn 4593  df-pr 4595  df-op 4599  df-opab 5177  df-xp 5670  df-rel 5671

This theorem is used by:  vvdifopab  38946