Theorem brabg 5486

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 5481	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴𝑅𝐵 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   class class class wbr 5097  {copab 5159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160

This theorem is referenced by:  brab  5490  ideqg  5799  brcnvg  5827  f1owe  7299  brrpssg  7670  soseq  8101  breng  8894  brdom2g  8896  brwdom  9474  brttrcl  9624  ltprord  10943  shftfib  14997  efgrelexlema  19680  isref  23455  sltval  27617  brsslt  27760  lrrecval  27919  istrkgld  28512  islnopp  28792  axcontlem5  29022  cmbr  31640  leopg  32178  cvbr  32338  mdbr  32350  dmdbr  32355  isfne  36512  brabg2  37887  isriscg  38154  brssr  38751  lcvbr  39316  bropabg  43602  nthrucw  47167