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Mirrors > Home > MPE Home > Th. List > rbropapd | Structured version Visualization version 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 5031 | . . . 4 ⊢ (𝐹𝑀𝑃 ↔ 〈𝐹, 𝑃〉 ∈ 𝑀) | |
2 | rbropapd.1 | . . . . 5 ⊢ (𝜑 → 𝑀 = {〈𝑓, 𝑝〉 ∣ (𝑓𝑊𝑝 ∧ 𝜓)}) | |
3 | 2 | eleq2d 2875 | . . . 4 ⊢ (𝜑 → (〈𝐹, 𝑃〉 ∈ 𝑀 ↔ 〈𝐹, 𝑃〉 ∈ {〈𝑓, 𝑝〉 ∣ (𝑓𝑊𝑝 ∧ 𝜓)})) |
4 | 1, 3 | syl5bb 286 | . . 3 ⊢ (𝜑 → (𝐹𝑀𝑃 ↔ 〈𝐹, 𝑃〉 ∈ {〈𝑓, 𝑝〉 ∣ (𝑓𝑊𝑝 ∧ 𝜓)})) |
5 | breq12 5035 | . . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝑓𝑊𝑝 ↔ 𝐹𝑊𝑃)) | |
6 | rbropapd.2 | . . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝜓 ↔ 𝜒)) | |
7 | 5, 6 | anbi12d 633 | . . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → ((𝑓𝑊𝑝 ∧ 𝜓) ↔ (𝐹𝑊𝑃 ∧ 𝜒))) |
8 | 7 | opelopabga 5385 | . . 3 ⊢ ((𝐹 ∈ 𝑋 ∧ 𝑃 ∈ 𝑌) → (〈𝐹, 𝑃〉 ∈ {〈𝑓, 𝑝〉 ∣ (𝑓𝑊𝑝 ∧ 𝜓)} ↔ (𝐹𝑊𝑃 ∧ 𝜒))) |
9 | 4, 8 | sylan9bb 513 | . 2 ⊢ ((𝜑 ∧ (𝐹 ∈ 𝑋 ∧ 𝑃 ∈ 𝑌)) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃 ∧ 𝜒))) |
10 | 9 | ex 416 | 1 ⊢ (𝜑 → ((𝐹 ∈ 𝑋 ∧ 𝑃 ∈ 𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃 ∧ 𝜒)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 〈cop 4531 class class class wbr 5030 {copab 5092 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 |
This theorem is referenced by: rbropap 5415 |
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