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Theorem eqrelf 37123
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 5785 . 2 ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑢𝑣(⟨𝑢, 𝑣⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐵)))
2 nfv 1918 . . 3 𝑢(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
3 nfv 1918 . . 3 𝑣(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
4 eqrelf.1 . . . . 5 𝑥𝐴
54nfel2 2922 . . . 4 𝑥𝑢, 𝑣⟩ ∈ 𝐴
6 eqrelf.2 . . . . 5 𝑥𝐵
76nfel2 2922 . . . 4 𝑥𝑢, 𝑣⟩ ∈ 𝐵
85, 7nfbi 1907 . . 3 𝑥(⟨𝑢, 𝑣⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐵)
9 eqrelf.3 . . . . 5 𝑦𝐴
109nfel2 2922 . . . 4 𝑦𝑢, 𝑣⟩ ∈ 𝐴
11 eqrelf.4 . . . . 5 𝑦𝐵
1211nfel2 2922 . . . 4 𝑦𝑢, 𝑣⟩ ∈ 𝐵
1310, 12nfbi 1907 . . 3 𝑦(⟨𝑢, 𝑣⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐵)
14 opeq12 4876 . . . . 5 ((𝑥 = 𝑢𝑦 = 𝑣) → ⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩)
1514eleq1d 2819 . . . 4 ((𝑥 = 𝑢𝑦 = 𝑣) → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐴))
1614eleq1d 2819 . . . 4 ((𝑥 = 𝑢𝑦 = 𝑣) → (⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐵))
1715, 16bibi12d 346 . . 3 ((𝑥 = 𝑢𝑦 = 𝑣) → ((⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ (⟨𝑢, 𝑣⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐵)))
182, 3, 8, 13, 17cbval2v 2340 . 2 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∀𝑢𝑣(⟨𝑢, 𝑣⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐵))
191, 18bitr4di 289 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 205  wa 397  wal 1540   = wceq 1542  wcel 2107  wnfc 2884  cop 4635  Rel wrel 5682
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 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-opab 5212  df-xp 5683  df-rel 5684
This theorem is referenced by:  vvdifopab  37128
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