Theorem eqrelf 34961

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 5509	. 2 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑢∀𝑣(⟨𝑢, 𝑣⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐵)))
2		nfv 1873	. . 3 ⊢ Ⅎ𝑢(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
3		nfv 1873	. . 3 ⊢ Ⅎ𝑣(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
4		eqrelf.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
5	4	nfel2 2948	. . . 4 ⊢ Ⅎ𝑥⟨𝑢, 𝑣⟩ ∈ 𝐴
6		eqrelf.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
7	6	nfel2 2948	. . . 4 ⊢ Ⅎ𝑥⟨𝑢, 𝑣⟩ ∈ 𝐵
8	5, 7	nfbi 1866	. . 3 ⊢ Ⅎ𝑥(⟨𝑢, 𝑣⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐵)
9		eqrelf.3	. . . . 5 ⊢ Ⅎ𝑦𝐴
10	9	nfel2 2948	. . . 4 ⊢ Ⅎ𝑦⟨𝑢, 𝑣⟩ ∈ 𝐴
11		eqrelf.4	. . . . 5 ⊢ Ⅎ𝑦𝐵
12	11	nfel2 2948	. . . 4 ⊢ Ⅎ𝑦⟨𝑢, 𝑣⟩ ∈ 𝐵
13	10, 12	nfbi 1866	. . 3 ⊢ Ⅎ𝑦(⟨𝑢, 𝑣⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐵)
14		opeq12 4680	. . . . 5 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩)
15	14	eleq1d 2850	. . . 4 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐴))
16	14	eleq1d 2850	. . . 4 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐵))
17	15, 16	bibi12d 338	. . 3 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ((⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ (⟨𝑢, 𝑣⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐵)))
18	2, 3, 8, 13, 17	cbval2 2345	. 2 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∀𝑢∀𝑣(⟨𝑢, 𝑣⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐵))
19	1, 18	syl6bbr 281	1 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387  ∀wal 1505   = wceq 1507   ∈ wcel 2050  Ⅎwnfc 2916  ⟨cop 4448  Rel wrel 5413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-rab 3097  df-v 3417  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-sn 4443  df-pr 4445  df-op 4449  df-opab 4993  df-xp 5414  df-rel 5415

This theorem is referenced by:  vvdifopab  34965