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Mirrors > Home > MPE Home > Th. List > Mathboxes > eqrelf | Structured version Visualization version GIF version |
Description: The equality connective between relations. (Contributed by Peter Mazsa, 25-Jun-2019.) |
Ref | Expression |
---|---|
eqrelf.1 | ⊢ Ⅎ𝑥𝐴 |
eqrelf.2 | ⊢ Ⅎ𝑥𝐵 |
eqrelf.3 | ⊢ Ⅎ𝑦𝐴 |
eqrelf.4 | ⊢ Ⅎ𝑦𝐵 |
Ref | Expression |
---|---|
eqrelf | ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqrel 5785 | . 2 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑢∀𝑣(⟨𝑢, 𝑣⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐵))) | |
2 | nfv 1918 | . . 3 ⊢ Ⅎ𝑢(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵) | |
3 | nfv 1918 | . . 3 ⊢ Ⅎ𝑣(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵) | |
4 | eqrelf.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
5 | 4 | nfel2 2922 | . . . 4 ⊢ Ⅎ𝑥⟨𝑢, 𝑣⟩ ∈ 𝐴 |
6 | eqrelf.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
7 | 6 | nfel2 2922 | . . . 4 ⊢ Ⅎ𝑥⟨𝑢, 𝑣⟩ ∈ 𝐵 |
8 | 5, 7 | nfbi 1907 | . . 3 ⊢ Ⅎ𝑥(⟨𝑢, 𝑣⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐵) |
9 | eqrelf.3 | . . . . 5 ⊢ Ⅎ𝑦𝐴 | |
10 | 9 | nfel2 2922 | . . . 4 ⊢ Ⅎ𝑦⟨𝑢, 𝑣⟩ ∈ 𝐴 |
11 | eqrelf.4 | . . . . 5 ⊢ Ⅎ𝑦𝐵 | |
12 | 11 | nfel2 2922 | . . . 4 ⊢ Ⅎ𝑦⟨𝑢, 𝑣⟩ ∈ 𝐵 |
13 | 10, 12 | nfbi 1907 | . . 3 ⊢ Ⅎ𝑦(⟨𝑢, 𝑣⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐵) |
14 | opeq12 4876 | . . . . 5 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩) | |
15 | 14 | eleq1d 2819 | . . . 4 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐴)) |
16 | 14 | eleq1d 2819 | . . . 4 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐵)) |
17 | 15, 16 | bibi12d 346 | . . 3 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ((⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ (⟨𝑢, 𝑣⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐵))) |
18 | 2, 3, 8, 13, 17 | cbval2v 2340 | . 2 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∀𝑢∀𝑣(⟨𝑢, 𝑣⟩ ∈ 𝐴 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝐵)) |
19 | 1, 18 | bitr4di 289 | 1 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(⟨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 Ⅎwnfc 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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