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Mirrors > Home > ILE Home > Th. List > rbropapd | GIF version |
Description: Properties of a pair in an extended binary relation. (Contributed by Alexander van der Vekens, 30-Oct-2017.) |
Ref | Expression |
---|---|
rbropapd.1 | ⊢ (𝜑 → 𝑀 = {〈𝑓, 𝑝〉 ∣ (𝑓𝑊𝑝 ∧ 𝜓)}) |
rbropapd.2 | ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
rbropapd | ⊢ (𝜑 → ((𝐹 ∈ 𝑋 ∧ 𝑃 ∈ 𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃 ∧ 𝜒)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 3930 | . . . 4 ⊢ (𝐹𝑀𝑃 ↔ 〈𝐹, 𝑃〉 ∈ 𝑀) | |
2 | rbropapd.1 | . . . . 5 ⊢ (𝜑 → 𝑀 = {〈𝑓, 𝑝〉 ∣ (𝑓𝑊𝑝 ∧ 𝜓)}) | |
3 | 2 | eleq2d 2209 | . . . 4 ⊢ (𝜑 → (〈𝐹, 𝑃〉 ∈ 𝑀 ↔ 〈𝐹, 𝑃〉 ∈ {〈𝑓, 𝑝〉 ∣ (𝑓𝑊𝑝 ∧ 𝜓)})) |
4 | 1, 3 | syl5bb 191 | . . 3 ⊢ (𝜑 → (𝐹𝑀𝑃 ↔ 〈𝐹, 𝑃〉 ∈ {〈𝑓, 𝑝〉 ∣ (𝑓𝑊𝑝 ∧ 𝜓)})) |
5 | breq12 3934 | . . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝑓𝑊𝑝 ↔ 𝐹𝑊𝑃)) | |
6 | rbropapd.2 | . . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝜓 ↔ 𝜒)) | |
7 | 5, 6 | anbi12d 464 | . . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → ((𝑓𝑊𝑝 ∧ 𝜓) ↔ (𝐹𝑊𝑃 ∧ 𝜒))) |
8 | 7 | opelopabga 4185 | . . 3 ⊢ ((𝐹 ∈ 𝑋 ∧ 𝑃 ∈ 𝑌) → (〈𝐹, 𝑃〉 ∈ {〈𝑓, 𝑝〉 ∣ (𝑓𝑊𝑝 ∧ 𝜓)} ↔ (𝐹𝑊𝑃 ∧ 𝜒))) |
9 | 4, 8 | sylan9bb 457 | . 2 ⊢ ((𝜑 ∧ (𝐹 ∈ 𝑋 ∧ 𝑃 ∈ 𝑌)) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃 ∧ 𝜒))) |
10 | 9 | ex 114 | 1 ⊢ (𝜑 → ((𝐹 ∈ 𝑋 ∧ 𝑃 ∈ 𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃 ∧ 𝜒)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 〈cop 3530 class class class wbr 3929 {copab 3988 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 |
This theorem is referenced by: rbropap 6140 |
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