Theorem brab 5131

Description: The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.)

Hypotheses

Ref	Expression
opelopab.1	⊢ 𝐴 ∈ V
opelopab.2	⊢ 𝐵 ∈ V
opelopab.3	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
opelopab.4	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
brab.5	⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Assertion

Ref	Expression
brab	⊢ (𝐴𝑅𝐵 ↔ 𝜒)

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

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

Proof of Theorem brab

Step	Hyp	Ref	Expression
1		opelopab.1	. 2 ⊢ 𝐴 ∈ V
2		opelopab.2	. 2 ⊢ 𝐵 ∈ V
3		opelopab.3	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4		opelopab.4	. . 3 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
5		brab.5	. . 3 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
6	3, 4, 5	brabg 5127	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑅𝐵 ↔ 𝜒))
7	1, 2, 6	mp2an 664	1 ⊢ (𝐴𝑅𝐵 ↔ 𝜒)

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1630   ∈ wcel 2144  Vcvv 3349   class class class wbr 4784  {copab 4844

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 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-opab 4845

This theorem is referenced by:  opbrop  5338  f1oweALT  7298  frxp  7437  fnwelem  7442  dftpos4  7522  dfac3  9143  axdc2lem  9471  brdom7disj  9554  brdom6disj  9555  ordpipq  9965  ltresr  10162  shftfn  14020  2shfti  14027  ishpg  25871  brcgr  26000  ex-opab  27625  br8d  29754  br8  31978  br6  31979  br4  31980  poseq  32084  dfbigcup2  32337  brsegle  32546  heiborlem2  33936