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Mirrors > Home > MPE Home > Th. List > 2rbropap | Structured version Visualization version GIF version |
Description: Properties of a pair in a restricted binary relation 𝑀 expressed as an ordered-pair class abstraction: 𝑀 is the binary relation 𝑊 restricted by the conditions 𝜓 and 𝜏. (Contributed by AV, 31-Jan-2021.) |
Ref | Expression |
---|---|
2rbropap.1 | ⊢ (𝜑 → 𝑀 = {〈𝑓, 𝑝〉 ∣ (𝑓𝑊𝑝 ∧ 𝜓 ∧ 𝜏)}) |
2rbropap.2 | ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝜓 ↔ 𝜒)) |
2rbropap.3 | ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝜏 ↔ 𝜃)) |
Ref | Expression |
---|---|
2rbropap | ⊢ ((𝜑 ∧ 𝐹 ∈ 𝑋 ∧ 𝑃 ∈ 𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃 ∧ 𝜒 ∧ 𝜃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2rbropap.1 | . . . 4 ⊢ (𝜑 → 𝑀 = {〈𝑓, 𝑝〉 ∣ (𝑓𝑊𝑝 ∧ 𝜓 ∧ 𝜏)}) | |
2 | 3anass 1095 | . . . . 5 ⊢ ((𝑓𝑊𝑝 ∧ 𝜓 ∧ 𝜏) ↔ (𝑓𝑊𝑝 ∧ (𝜓 ∧ 𝜏))) | |
3 | 2 | opabbii 5172 | . . . 4 ⊢ {〈𝑓, 𝑝〉 ∣ (𝑓𝑊𝑝 ∧ 𝜓 ∧ 𝜏)} = {〈𝑓, 𝑝〉 ∣ (𝑓𝑊𝑝 ∧ (𝜓 ∧ 𝜏))} |
4 | 1, 3 | eqtrdi 2792 | . . 3 ⊢ (𝜑 → 𝑀 = {〈𝑓, 𝑝〉 ∣ (𝑓𝑊𝑝 ∧ (𝜓 ∧ 𝜏))}) |
5 | 2rbropap.2 | . . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝜓 ↔ 𝜒)) | |
6 | 2rbropap.3 | . . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝜏 ↔ 𝜃)) | |
7 | 5, 6 | anbi12d 631 | . . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → ((𝜓 ∧ 𝜏) ↔ (𝜒 ∧ 𝜃))) |
8 | 4, 7 | rbropap 5522 | . 2 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝑋 ∧ 𝑃 ∈ 𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃 ∧ (𝜒 ∧ 𝜃)))) |
9 | 3anass 1095 | . 2 ⊢ ((𝐹𝑊𝑃 ∧ 𝜒 ∧ 𝜃) ↔ (𝐹𝑊𝑃 ∧ (𝜒 ∧ 𝜃))) | |
10 | 8, 9 | bitr4di 288 | 1 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝑋 ∧ 𝑃 ∈ 𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃 ∧ 𝜒 ∧ 𝜃))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 class class class wbr 5105 {copab 5167 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-br 5106 df-opab 5168 |
This theorem is referenced by: iswlkon 28605 |
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