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| Mirrors > Home > MPE Home > Th. List > eqrel | Structured version Visualization version GIF version | ||
| Description: Extensionality principle for relations. Theorem 3.2(ii) of [Monk1] p. 33. (Contributed by NM, 2-Aug-1994.) |
| Ref | Expression |
|---|---|
| eqrel | ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ssrel 5792 | . . 3 ⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 〈𝑥, 𝑦〉 ∈ 𝐵))) | |
| 2 | ssrel 5792 | . . 3 ⊢ (Rel 𝐵 → (𝐵 ⊆ 𝐴 ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐵 → 〈𝑥, 𝑦〉 ∈ 𝐴))) | |
| 3 | 1, 2 | bi2anan9 638 | . 2 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴) ↔ (∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 〈𝑥, 𝑦〉 ∈ 𝐵) ∧ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐵 → 〈𝑥, 𝑦〉 ∈ 𝐴)))) |
| 4 | eqss 3999 | . 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)) | |
| 5 | 2albiim 1890 | . 2 ⊢ (∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵) ↔ (∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 〈𝑥, 𝑦〉 ∈ 𝐵) ∧ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐵 → 〈𝑥, 𝑦〉 ∈ 𝐴))) | |
| 6 | 3, 4, 5 | 3bitr4g 314 | 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 206 ∧ wa 395 ∀wal 1538 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 〈cop 4632 Rel wrel 5690 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-ss 3968 df-opab 5206 df-xp 5691 df-rel 5692 |
| This theorem is referenced by: eqrelriv 5799 eqrelrdv 5802 eqbrrdv 5803 eqrelrdv2 5805 opabid2 5838 reldm0 5938 iss 6053 asymref 6136 funssres 6610 fsn 7155 eqrelf 38256 iss2 38345 |
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