Theorem isprmpt2 5858

Description: Properties of a pair in an extended binary relation. (Contributed by Alexander van der Vekens, 30-Oct-2017.)

Hypotheses

Ref	Expression
isprmpt2.1	⊢ (𝜑 → 𝑀 = {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝 ∧ 𝜓)})
isprmpt2.2	⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝜓 ↔ 𝜒))

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

Step	Hyp	Ref	Expression
1		df-br 3765	. . . 4 ⊢ (𝐹𝑀𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ 𝑀)
2		isprmpt2.1	. . . . . 6 ⊢ (𝜑 → 𝑀 = {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝 ∧ 𝜓)})
3	2	adantr 261	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ 𝑋 ∧ 𝑃 ∈ 𝑌)) → 𝑀 = {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝 ∧ 𝜓)})
4	3	eleq2d 2107	. . . 4 ⊢ ((𝜑 ∧ (𝐹 ∈ 𝑋 ∧ 𝑃 ∈ 𝑌)) → (⟨𝐹, 𝑃⟩ ∈ 𝑀 ↔ ⟨𝐹, 𝑃⟩ ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝 ∧ 𝜓)}))
5	1, 4	syl5bb 181	. . 3 ⊢ ((𝜑 ∧ (𝐹 ∈ 𝑋 ∧ 𝑃 ∈ 𝑌)) → (𝐹𝑀𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝 ∧ 𝜓)}))
6		breq12 3769	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝑓𝑊𝑝 ↔ 𝐹𝑊𝑃))
7		isprmpt2.2	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝜓 ↔ 𝜒))
8	6, 7	anbi12d 442	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → ((𝑓𝑊𝑝 ∧ 𝜓) ↔ (𝐹𝑊𝑃 ∧ 𝜒)))
9	8	opelopabga 4000	. . . 4 ⊢ ((𝐹 ∈ 𝑋 ∧ 𝑃 ∈ 𝑌) → (⟨𝐹, 𝑃⟩ ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝 ∧ 𝜓)} ↔ (𝐹𝑊𝑃 ∧ 𝜒)))
10	9	adantl 262	. . 3 ⊢ ((𝜑 ∧ (𝐹 ∈ 𝑋 ∧ 𝑃 ∈ 𝑌)) → (⟨𝐹, 𝑃⟩ ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝 ∧ 𝜓)} ↔ (𝐹𝑊𝑃 ∧ 𝜒)))
11	5, 10	bitrd 177	. 2 ⊢ ((𝜑 ∧ (𝐹 ∈ 𝑋 ∧ 𝑃 ∈ 𝑌)) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃 ∧ 𝜒)))
12	11	ex 108	1 ⊢ (𝜑 → ((𝐹 ∈ 𝑋 ∧ 𝑃 ∈ 𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃 ∧ 𝜒))))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393  ⟨cop 3378   class class class wbr 3764  {copab 3817

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819

This theorem is referenced by: (None)