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Mirrors > Home > MPE Home > Th. List > Mathboxes > brabd0 | 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 |
---|---|
brabd0.x | ⊢ (𝜑 → ∀𝑥𝜑) |
brabd0.y | ⊢ (𝜑 → ∀𝑦𝜑) |
brabd0.xch | ⊢ (𝜑 → Ⅎ𝑥𝜒) |
brabd0.ych | ⊢ (𝜑 → Ⅎ𝑦𝜒) |
brabd0.exa | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
brabd0.exb | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
brabd0.def | ⊢ (𝜑 → 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜓}) |
brabd0.is | ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
brabd0 | ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 5040 | . . 3 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
2 | brabd0.def | . . . 4 ⊢ (𝜑 → 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜓}) | |
3 | 2 | eleq2d 2816 | . . 3 ⊢ (𝜑 → (〈𝐴, 𝐵〉 ∈ 𝑅 ↔ 〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓})) |
4 | 1, 3 | syl5bb 286 | . 2 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓})) |
5 | brabd0.x | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
6 | brabd0.y | . . 3 ⊢ (𝜑 → ∀𝑦𝜑) | |
7 | brabd0.xch | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
8 | brabd0.ych | . . 3 ⊢ (𝜑 → Ⅎ𝑦𝜒) | |
9 | brabd0.exa | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
10 | brabd0.exb | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
11 | brabd0.is | . . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒)) | |
12 | 5, 6, 7, 8, 9, 10, 11 | opelopabd 34996 | . 2 ⊢ (𝜑 → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓} ↔ 𝜒)) |
13 | 4, 12 | bitrd 282 | 1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1541 = wceq 1543 Ⅎwnf 1791 ∈ wcel 2112 〈cop 4533 class class class wbr 5039 {copab 5101 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 |
This theorem is referenced by: brabd 35003 |
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