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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 5067 | . . 3 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
2 | brabd0.def | . . . 4 ⊢ (𝜑 → 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜓}) | |
3 | 2 | eleq2d 2898 | . . 3 ⊢ (𝜑 → (〈𝐴, 𝐵〉 ∈ 𝑅 ↔ 〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓})) |
4 | 1, 3 | syl5bb 285 | . 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 34436 | . 2 ⊢ (𝜑 → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓} ↔ 𝜒)) |
13 | 4, 12 | bitrd 281 | 1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1535 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 〈cop 4573 class class class wbr 5066 {copab 5128 |
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 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 |
This theorem is referenced by: brabd 34443 |
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