Theorem brres 5987

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 5986	. 2 ⊢ (𝐶 ∈ 𝑉 → (⟨𝐵, 𝐶⟩ ∈ (𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝑅)))
2		df-br 5112	. 2 ⊢ (𝐵(𝑅 ↾ 𝐴)𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ (𝑅 ↾ 𝐴))
3		df-br 5112	. . 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 2146  ⟨cop 4597   class class class wbr 5111   ↾ cres 5665

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-xp 5669  df-res 5675

This theorem is used by:  brresi  5989  dfima2  6066  predres  6344  elecres  8749  ttrclselem2  9702  axhcompl-zf  31421  fv1stcnv  36306  fv2ndcnv  36307  bj-idreseq  37863  bj-idreseqb  37864  brcnvepres  38979  brres2  38980  eldmres  38984  elrnres  38985  brinxprnres  39004  exanres  39008  eqres  39047  alrmomorn  39065  alrmomodm  39066  brxrn  39090  rnxrnres  39129  1cossres  39226  brressn  39238  eldm1cossres  39257  brssrres  39291  disjres  39551  antisymrelres  39573  dfdfat2  47923