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Theorem brab2a 4530
Description: Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 9-Nov-2015.)
Hypotheses
Ref Expression
brab2a.1 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
brab2a.2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)}
Assertion
Ref Expression
brab2a (𝐴𝑅𝐵 ↔ ((𝐴𝐶𝐵𝐷) ∧ 𝜓))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦   𝜓,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑅(𝑥,𝑦)

Proof of Theorem brab2a
StepHypRef Expression
1 simpl 108 . . . . 5 (((𝑥𝐶𝑦𝐷) ∧ 𝜑) → (𝑥𝐶𝑦𝐷))
21ssopab2i 4137 . . . 4 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦𝐷)}
3 brab2a.2 . . . 4 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)}
4 df-xp 4483 . . . 4 (𝐶 × 𝐷) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦𝐷)}
52, 3, 43sstr4i 3088 . . 3 𝑅 ⊆ (𝐶 × 𝐷)
65brel 4529 . 2 (𝐴𝑅𝐵 → (𝐴𝐶𝐵𝐷))
7 df-br 3876 . . . 4 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
83eleq2i 2166 . . . 4 (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)})
97, 8bitri 183 . . 3 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)})
10 brab2a.1 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
1110opelopab2a 4125 . . 3 ((𝐴𝐶𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)} ↔ 𝜓))
129, 11syl5bb 191 . 2 ((𝐴𝐶𝐵𝐷) → (𝐴𝑅𝐵𝜓))
136, 12biadan2 447 1 (𝐴𝑅𝐵 ↔ ((𝐴𝐶𝐵𝐷) ∧ 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1299  wcel 1448  cop 3477   class class class wbr 3875  {copab 3928   × cxp 4475
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-br 3876  df-opab 3930  df-xp 4483
This theorem is referenced by:  lmbr  12163
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