Theorem brabg 5499

Description: The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 19-Dec-2013.)

Hypotheses

Ref	Expression
opelopabg.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
opelopabg.2	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
brabg.5	⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Assertion

Ref	Expression
brabg	⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴𝑅𝐵 ↔ 𝜒))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜒,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑅(𝑥,𝑦)

Proof of Theorem brabg

Step	Hyp	Ref	Expression
1		opelopabg.1	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
2		opelopabg.2	. . 3 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
3	1, 2	sylan9bb 509	. 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜒))
4		brabg.5	. 2 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
5	3, 4	brabga 5494	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴𝑅𝐵 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   class class class wbr 5107  {copab 5169

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170

This theorem is referenced by:  brab  5503  ideqg  5815  brcnvg  5843  f1owe  7328  brrpssg  7701  soseq  8138  breng  8927  brdom2g  8929  brwdom  9520  brttrcl  9666  ltprord  10983  shftfib  15038  efgrelexlema  19679  isref  23396  sltval  27559  brsslt  27697  lrrecval  27846  istrkgld  28386  islnopp  28666  axcontlem5  28895  cmbr  31513  leopg  32051  cvbr  32211  mdbr  32223  dmdbr  32228  isfne  36327  brabg2  37711  isriscg  37978  brssr  38492  lcvbr  39014  bropabg  43312