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 5103	. 2 ⊢ (𝐵(𝑅 ↾ 𝐴)𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ (𝑅 ↾ 𝐴))
3		df-br 5103	. . 3 ⊢ (𝐵𝑅𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝑅)
4	3	anbi2i 635	. 2 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝐶) ↔ (𝐵 ∈ 𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝑅))
5	1, 2, 4	3bitr4g 317	1 ⊢ (𝐶 ∈ 𝑉 → (𝐵(𝑅 ↾ 𝐴)𝐶 ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝐶)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2144  ⟨cop 4834   class class class wbr 5102   ↾ cres 5653

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5657  df-res 5663

This theorem is used by:  brresi  5980  dfima2  6060  predres  6345  elecres  8774  ttrclselem2  9734  axhcompl-zf  31609  fv1stcnv  36665  fv2ndcnv  36666  bj-idreseq  38220  bj-idreseqb  38221  brcnvepres  39341  brres2  39342  eldmres  39346  elrnres  39347  brinxprnres  39366  exanres  39370  eqres  39409  alrmomorn  39427  alrmomodm  39428  brxrn  39452  rnxrnres  39491  1cossres  39588  brressn  39600  eldm1cossres  39619  brssrres  39653  disjres  39913  antisymrelres  39935  dfdfat2  48334