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Theorem isprmpt2 5888
Description: Properties of a pair in an extended binary relation. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
Hypotheses
Ref Expression
isprmpt2.1 (𝜑𝑀 = {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝𝜓)})
isprmpt2.2 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝜓𝜒))
Assertion
Ref Expression
isprmpt2 (𝜑 → ((𝐹𝑋𝑃𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃𝜒))))
Distinct variable groups:   𝑓,𝐹,𝑝   𝑃,𝑓,𝑝   𝑓,𝑊,𝑝   𝜒,𝑓,𝑝
Allowed substitution hints:   𝜑(𝑓,𝑝)   𝜓(𝑓,𝑝)   𝑀(𝑓,𝑝)   𝑋(𝑓,𝑝)   𝑌(𝑓,𝑝)

Proof of Theorem isprmpt2
StepHypRef Expression
1 df-br 3792 . . . 4 (𝐹𝑀𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ 𝑀)
2 isprmpt2.1 . . . . . 6 (𝜑𝑀 = {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝𝜓)})
32adantr 265 . . . . 5 ((𝜑 ∧ (𝐹𝑋𝑃𝑌)) → 𝑀 = {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝𝜓)})
43eleq2d 2123 . . . 4 ((𝜑 ∧ (𝐹𝑋𝑃𝑌)) → (⟨𝐹, 𝑃⟩ ∈ 𝑀 ↔ ⟨𝐹, 𝑃⟩ ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝𝜓)}))
51, 4syl5bb 185 . . 3 ((𝜑 ∧ (𝐹𝑋𝑃𝑌)) → (𝐹𝑀𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝𝜓)}))
6 breq12 3796 . . . . . 6 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑓𝑊𝑝𝐹𝑊𝑃))
7 isprmpt2.2 . . . . . 6 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝜓𝜒))
86, 7anbi12d 450 . . . . 5 ((𝑓 = 𝐹𝑝 = 𝑃) → ((𝑓𝑊𝑝𝜓) ↔ (𝐹𝑊𝑃𝜒)))
98opelopabga 4027 . . . 4 ((𝐹𝑋𝑃𝑌) → (⟨𝐹, 𝑃⟩ ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝𝜓)} ↔ (𝐹𝑊𝑃𝜒)))
109adantl 266 . . 3 ((𝜑 ∧ (𝐹𝑋𝑃𝑌)) → (⟨𝐹, 𝑃⟩ ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝𝜓)} ↔ (𝐹𝑊𝑃𝜒)))
115, 10bitrd 181 . 2 ((𝜑 ∧ (𝐹𝑋𝑃𝑌)) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃𝜒)))
1211ex 112 1 (𝜑 → ((𝐹𝑋𝑃𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃𝜒))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  cop 3405   class class class wbr 3791  {copab 3844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846
This theorem is referenced by: (None)
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