Theorem brres 5946

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 5945	. 2 ⊢ (𝐶 ∈ 𝑉 → (⟨𝐵, 𝐶⟩ ∈ (𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝑅)))
2		df-br 5087	. 2 ⊢ (𝐵(𝑅 ↾ 𝐴)𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ (𝑅 ↾ 𝐴))
3		df-br 5087	. . 3 ⊢ (𝐵𝑅𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝑅)
4	3	anbi2i 624	. 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 206   ∧ wa 395   ∈ wcel 2114  ⟨cop 4574   class class class wbr 5086   ↾ cres 5627

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 2709  ax-sep 5232  ax-pr 5371

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 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5631  df-res 5637

This theorem is referenced by:  brresi  5948  dfima2  6022  predres  6298  elecres  8686  ttrclselem2  9641  axhcompl-zf  31087  fv1stcnv  35978  fv2ndcnv  35979  bj-idreseq  37495  bj-idreseqb  37496  brcnvepres  38610  brres2  38611  eldmres  38615  elrnres  38616  brinxprnres  38635  exanres  38639  eqres  38678  alrmomorn  38696  alrmomodm  38697  brxrn  38721  rnxrnres  38760  1cossres  38857  brressn  38869  eldm1cossres  38888  brssrres  38922  disjres  39182  antisymrelres  39204  dfdfat2  47591