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Mirrors > Home > MPE Home > Th. List > Mathboxes > brabd | Structured version Visualization version GIF version |
Description: Expressing that two sets are related by a binary relation which is expressed as a class abstraction of ordered pairs. (Contributed by BJ, 17-Dec-2023.) |
Ref | Expression |
---|---|
brabd.exa | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
brabd.exb | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
brabd.def | ⊢ (𝜑 → 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜓}) |
brabd.is | ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
brabd | ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-5 1907 | . 2 ⊢ (𝜑 → ∀𝑥𝜑) | |
2 | ax-5 1907 | . 2 ⊢ (𝜑 → ∀𝑦𝜑) | |
3 | nfvd 1912 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
4 | nfvd 1912 | . 2 ⊢ (𝜑 → Ⅎ𝑦𝜒) | |
5 | brabd.exa | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
6 | brabd.exb | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
7 | brabd.def | . 2 ⊢ (𝜑 → 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜓}) | |
8 | brabd.is | . 2 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒)) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | brabd0 37129 | 1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 class class class wbr 5147 {copab 5209 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 |
This theorem is referenced by: bj-imdirval3 37166 |
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