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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 1910 | . 2 ⊢ (𝜑 → ∀𝑥𝜑) | |
| 2 | ax-5 1910 | . 2 ⊢ (𝜑 → ∀𝑦𝜑) | |
| 3 | nfvd 1915 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
| 4 | nfvd 1915 | . 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 37148 | 1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝜒)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 {copab 5205 |
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 |
| This theorem is referenced by: bj-imdirval3 37185 |
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