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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 5658 | . 2 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑢∀𝑣(〈𝑢, 𝑣〉 ∈ 𝐴 ↔ 〈𝑢, 𝑣〉 ∈ 𝐵))) | |
2 | nfv 1915 | . . 3 ⊢ Ⅎ𝑢(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵) | |
3 | nfv 1915 | . . 3 ⊢ Ⅎ𝑣(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵) | |
4 | eqrelf.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
5 | 4 | nfel2 2996 | . . . 4 ⊢ Ⅎ𝑥〈𝑢, 𝑣〉 ∈ 𝐴 |
6 | eqrelf.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
7 | 6 | nfel2 2996 | . . . 4 ⊢ Ⅎ𝑥〈𝑢, 𝑣〉 ∈ 𝐵 |
8 | 5, 7 | nfbi 1904 | . . 3 ⊢ Ⅎ𝑥(〈𝑢, 𝑣〉 ∈ 𝐴 ↔ 〈𝑢, 𝑣〉 ∈ 𝐵) |
9 | eqrelf.3 | . . . . 5 ⊢ Ⅎ𝑦𝐴 | |
10 | 9 | nfel2 2996 | . . . 4 ⊢ Ⅎ𝑦〈𝑢, 𝑣〉 ∈ 𝐴 |
11 | eqrelf.4 | . . . . 5 ⊢ Ⅎ𝑦𝐵 | |
12 | 11 | nfel2 2996 | . . . 4 ⊢ Ⅎ𝑦〈𝑢, 𝑣〉 ∈ 𝐵 |
13 | 10, 12 | nfbi 1904 | . . 3 ⊢ Ⅎ𝑦(〈𝑢, 𝑣〉 ∈ 𝐴 ↔ 〈𝑢, 𝑣〉 ∈ 𝐵) |
14 | opeq12 4805 | . . . . 5 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 〈𝑥, 𝑦〉 = 〈𝑢, 𝑣〉) | |
15 | 14 | eleq1d 2897 | . . . 4 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑢, 𝑣〉 ∈ 𝐴)) |
16 | 14 | eleq1d 2897 | . . . 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 2363 | . 2 ⊢ (∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵) ↔ ∀𝑢∀𝑣(〈𝑢, 𝑣〉 ∈ 𝐴 ↔ 〈𝑢, 𝑣〉 ∈ 𝐵)) |
19 | 1, 18 | syl6bbr 291 | 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 208 ∧ wa 398 ∀wal 1535 = wceq 1537 ∈ wcel 2114 Ⅎwnfc 2961 〈cop 4573 Rel wrel 5560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-opab 5129 df-xp 5561 df-rel 5562 |
This theorem is referenced by: vvdifopab 35536 |
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