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Theorem brabgaf 29725
Description: The law of concretion for a binary relation. (Contributed by Mario Carneiro, 19-Dec-2013.) (Revised by Thierry Arnoux, 17-May-2020.)
Hypotheses
Ref Expression
brabgaf.0 𝑥𝜓
brabgaf.1 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
brabgaf.2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Assertion
Ref Expression
brabgaf ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵𝜓))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜓,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥)   𝑅(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem brabgaf
StepHypRef Expression
1 df-br 4803 . . 3 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2 brabgaf.2 . . . 4 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
32eleq2i 2829 . . 3 (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
41, 3bitri 264 . 2 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
5 elopab 5131 . . 3 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
6 elisset 3353 . . . 4 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
7 elisset 3353 . . . 4 (𝐵𝑊 → ∃𝑦 𝑦 = 𝐵)
8 eeanv 2325 . . . . 5 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
9 nfe1 2174 . . . . . . 7 𝑥𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
10 brabgaf.0 . . . . . . 7 𝑥𝜓
119, 10nfbi 1980 . . . . . 6 𝑥(∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓)
12 nfe1 2174 . . . . . . . . 9 𝑦𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
1312nfex 2299 . . . . . . . 8 𝑦𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
14 nfv 1990 . . . . . . . 8 𝑦𝜓
1513, 14nfbi 1980 . . . . . . 7 𝑦(∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓)
16 opeq12 4553 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵) → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
17 copsexg 5102 . . . . . . . . . 10 (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
1817eqcoms 2766 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ → (𝜑 ↔ ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
1916, 18syl 17 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑 ↔ ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
20 brabgaf.1 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
2119, 20bitr3d 270 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓))
2215, 21exlimi 2231 . . . . . 6 (∃𝑦(𝑥 = 𝐴𝑦 = 𝐵) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓))
2311, 22exlimi 2231 . . . . 5 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓))
248, 23sylbir 225 . . . 4 ((∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓))
256, 7, 24syl2an 495 . . 3 ((𝐴𝑉𝐵𝑊) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓))
265, 25syl5bb 272 . 2 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜓))
274, 26syl5bb 272 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1630  wex 1851  wnf 1855  wcel 2137  cop 4325   class class class wbr 4802  {copab 4862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pr 5053
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-rab 3057  df-v 3340  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-br 4803  df-opab 4863
This theorem is referenced by:  fmptcof2  29764
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