Theorem brres 5978

Description: Binary relation on a restriction. (Contributed by Mario Carneiro, 4-Nov-2015.) Commute the consequent. (Revised by Peter Mazsa, 24-Sep-2022.)

Assertion

Ref	Expression
brres	⊢ (𝐶 ∈ 𝑉 → (𝐵(𝑅 ↾ 𝐴)𝐶 ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝐶)))

Proof of Theorem brres

Step	Hyp	Ref	Expression
1		opelres 5977	. 2 ⊢ (𝐶 ∈ 𝑉 → (⟨𝐵, 𝐶⟩ ∈ (𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝑅)))
2		df-br 5139	. 2 ⊢ (𝐵(𝑅 ↾ 𝐴)𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ (𝑅 ↾ 𝐴))
3		df-br 5139	. . 3 ⊢ (𝐵𝑅𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝑅)
4	3	anbi2i 622	. 2 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝐶) ↔ (𝐵 ∈ 𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝑅))
5	1, 2, 4	3bitr4g 314	1 ⊢ (𝐶 ∈ 𝑉 → (𝐵(𝑅 ↾ 𝐴)𝐶 ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝐶)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∈ wcel 2098  ⟨cop 4626   class class class wbr 5138   ↾ cres 5668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-xp 5672  df-res 5678

This theorem is referenced by:  brresi  5980  dfima2  6051  predres  6330  ttrclselem2  9717  axhcompl-zf  30720  fv1stcnv  35243  fv2ndcnv  35244  bj-idreseq  36533  bj-idreseqb  36534  brcnvepres  37625  brres2  37626  eldmres  37628  elrnres  37629  elecres  37635  brinxprnres  37650  exanres  37654  eqres  37699  alrmomorn  37717  alrmomodm  37718  brxrn  37734  rnxrnres  37759  1cossres  37789  brressn  37801  eldm1cossres  37820  brssrres  37864  disjres  38104  antisymrelres  38123  dfdfat2  46321