Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > eqrel2 | Structured version Visualization version GIF version |
Description: Equality of relations. (Contributed by Peter Mazsa, 8-Mar-2019.) |
Ref | Expression |
---|---|
eqrel2 | ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrel3 35571 | . . 3 ⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(𝑥𝐴𝑦 → 𝑥𝐵𝑦))) | |
2 | ssrel3 35571 | . . 3 ⊢ (Rel 𝐵 → (𝐵 ⊆ 𝐴 ↔ ∀𝑥∀𝑦(𝑥𝐵𝑦 → 𝑥𝐴𝑦))) | |
3 | 1, 2 | bi2anan9 637 | . 2 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴) ↔ (∀𝑥∀𝑦(𝑥𝐴𝑦 → 𝑥𝐵𝑦) ∧ ∀𝑥∀𝑦(𝑥𝐵𝑦 → 𝑥𝐴𝑦)))) |
4 | eqss 3982 | . 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)) | |
5 | 2albiim 1891 | . 2 ⊢ (∀𝑥∀𝑦(𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦) ↔ (∀𝑥∀𝑦(𝑥𝐴𝑦 → 𝑥𝐵𝑦) ∧ ∀𝑥∀𝑦(𝑥𝐵𝑦 → 𝑥𝐴𝑦))) | |
6 | 3, 4, 5 | 3bitr4g 316 | 1 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1535 = wceq 1537 ⊆ wss 3936 class class class wbr 5066 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-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-in 3943 df-ss 3952 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |