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Theorem eqrelf 34344
Description: The equality connective between relations. (Contributed by Peter Mazsa, 25-Jun-2019.)
Hypotheses
Ref Expression
eqrelf.1 𝑥𝐴
eqrelf.2 𝑥𝐵
eqrelf.3 𝑦𝐴
eqrelf.4 𝑦𝐵
Assertion
Ref Expression
eqrelf ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥𝑦(⟨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.
StepHypRef Expression
1 eqrel 5366 . 2 ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑢𝑣(⟨𝑢, 𝑣⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐵)))
2 nfv 1992 . . 3 𝑢(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
3 nfv 1992 . . 3 𝑣(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
4 eqrelf.1 . . . . 5 𝑥𝐴
54nfel2 2919 . . . 4 𝑥𝑢, 𝑣⟩ ∈ 𝐴
6 eqrelf.2 . . . . 5 𝑥𝐵
76nfel2 2919 . . . 4 𝑥𝑢, 𝑣⟩ ∈ 𝐵
85, 7nfbi 1982 . . 3 𝑥(⟨𝑢, 𝑣⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐵)
9 eqrelf.3 . . . . 5 𝑦𝐴
109nfel2 2919 . . . 4 𝑦𝑢, 𝑣⟩ ∈ 𝐴
11 eqrelf.4 . . . . 5 𝑦𝐵
1211nfel2 2919 . . . 4 𝑦𝑢, 𝑣⟩ ∈ 𝐵
1310, 12nfbi 1982 . . 3 𝑦(⟨𝑢, 𝑣⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐵)
14 opeq12 4555 . . . . 5 ((𝑥 = 𝑢𝑦 = 𝑣) → ⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩)
1514eleq1d 2824 . . . 4 ((𝑥 = 𝑢𝑦 = 𝑣) → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐴))
1614eleq1d 2824 . . . 4 ((𝑥 = 𝑢𝑦 = 𝑣) → (⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐵))
1715, 16bibi12d 334 . . 3 ((𝑥 = 𝑢𝑦 = 𝑣) → ((⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ (⟨𝑢, 𝑣⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐵)))
182, 3, 8, 13, 17cbval2 2424 . 2 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∀𝑢𝑣(⟨𝑢, 𝑣⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐵))
191, 18syl6bbr 278 1 ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1630   = wceq 1632  wcel 2139  wnfc 2889  cop 4327  Rel wrel 5271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-opab 4865  df-xp 5272  df-rel 5273
This theorem is referenced by:  vvdifopab  34348
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