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Theorem brab2a 5091
 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 472 . . . . 5 (((𝑥𝐶𝑦𝐷) ∧ 𝜑) → (𝑥𝐶𝑦𝐷))
21ssopab2i 4928 . . . 4 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦𝐷)}
3 brab2a.2 . . . 4 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)}
4 df-xp 5044 . . . 4 (𝐶 × 𝐷) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦𝐷)}
52, 3, 43sstr4i 3607 . . 3 𝑅 ⊆ (𝐶 × 𝐷)
65brel 5090 . 2 (𝐴𝑅𝐵 → (𝐴𝐶𝐵𝐷))
7 df-br 4584 . . . 4 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
83eleq2i 2680 . . . 4 (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)})
97, 8bitri 263 . . 3 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)})
10 brab2a.1 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
1110opelopab2a 4915 . . 3 ((𝐴𝐶𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)} ↔ 𝜓))
129, 11syl5bb 271 . 2 ((𝐴𝐶𝐵𝐷) → (𝐴𝑅𝐵𝜓))
136, 12biadan2 672 1 (𝐴𝑅𝐵 ↔ ((𝐴𝐶𝐵𝐷) ∧ 𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ⟨cop 4131   class class class wbr 4583  {copab 4642   × cxp 5036 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044 This theorem is referenced by:  issect  16236  iscgrg  25207  ishlg  25297  iscgra  25501  isinag  25529  isleag  25533
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