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Theorem isprmpt2 5776
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⟩ ∣ (f𝑊𝑝 ψ)})
isprmpt2.2 ((f = 𝐹 𝑝 = 𝑃) → (ψχ))
Assertion
Ref Expression
isprmpt2 (φ → ((𝐹 𝑋 𝑃 𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃 χ))))
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 3735 . . . 4 (𝐹𝑀𝑃 ↔ ⟨𝐹, 𝑃 𝑀)
2 isprmpt2.1 . . . . . 6 (φ𝑀 = {⟨f, 𝑝⟩ ∣ (f𝑊𝑝 ψ)})
32adantr 261 . . . . 5 ((φ (𝐹 𝑋 𝑃 𝑌)) → 𝑀 = {⟨f, 𝑝⟩ ∣ (f𝑊𝑝 ψ)})
43eleq2d 2085 . . . 4 ((φ (𝐹 𝑋 𝑃 𝑌)) → (⟨𝐹, 𝑃 𝑀 ↔ ⟨𝐹, 𝑃 {⟨f, 𝑝⟩ ∣ (f𝑊𝑝 ψ)}))
51, 4syl5bb 181 . . 3 ((φ (𝐹 𝑋 𝑃 𝑌)) → (𝐹𝑀𝑃 ↔ ⟨𝐹, 𝑃 {⟨f, 𝑝⟩ ∣ (f𝑊𝑝 ψ)}))
6 breq12 3739 . . . . . 6 ((f = 𝐹 𝑝 = 𝑃) → (f𝑊𝑝𝐹𝑊𝑃))
7 isprmpt2.2 . . . . . 6 ((f = 𝐹 𝑝 = 𝑃) → (ψχ))
86, 7anbi12d 445 . . . . 5 ((f = 𝐹 𝑝 = 𝑃) → ((f𝑊𝑝 ψ) ↔ (𝐹𝑊𝑃 χ)))
98opelopabga 3970 . . . 4 ((𝐹 𝑋 𝑃 𝑌) → (⟨𝐹, 𝑃 {⟨f, 𝑝⟩ ∣ (f𝑊𝑝 ψ)} ↔ (𝐹𝑊𝑃 χ)))
109adantl 262 . . 3 ((φ (𝐹 𝑋 𝑃 𝑌)) → (⟨𝐹, 𝑃 {⟨f, 𝑝⟩ ∣ (f𝑊𝑝 ψ)} ↔ (𝐹𝑊𝑃 χ)))
115, 10bitrd 177 . 2 ((φ (𝐹 𝑋 𝑃 𝑌)) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃 χ)))
1211ex 108 1 (φ → ((𝐹 𝑋 𝑃 𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃 χ))))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1226   wcel 1370  cop 3349   class class class wbr 3734  {copab 3787
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789
This theorem is referenced by: (None)
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