Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  brres2 Structured version   Visualization version   GIF version

Theorem brres2 34467
Description: Binary relation on a restriction. (Contributed by Peter Mazsa, 2-Jan-2019.) (Revised by Peter Mazsa, 16-Dec-2021.)
Assertion
Ref Expression
brres2 (𝐵(𝑅𝐴)𝐶𝐵(𝑅 ∩ (𝐴 × ran (𝑅𝐴)))𝐶)

Proof of Theorem brres2
StepHypRef Expression
1 brres 5572 . . 3 (𝐶 ∈ ran (𝑅𝐴) → (𝐵(𝑅𝐴)𝐶 ↔ (𝐵𝐴𝐵𝑅𝐶)))
21pm5.32i 570 . 2 ((𝐶 ∈ ran (𝑅𝐴) ∧ 𝐵(𝑅𝐴)𝐶) ↔ (𝐶 ∈ ran (𝑅𝐴) ∧ (𝐵𝐴𝐵𝑅𝐶)))
3 relres 5601 . . . 4 Rel (𝑅𝐴)
43relelrni 5532 . . 3 (𝐵(𝑅𝐴)𝐶𝐶 ∈ ran (𝑅𝐴))
54pm4.71ri 556 . 2 (𝐵(𝑅𝐴)𝐶 ↔ (𝐶 ∈ ran (𝑅𝐴) ∧ 𝐵(𝑅𝐴)𝐶))
6 brinxp2 5348 . . 3 (𝐵(𝑅 ∩ (𝐴 × ran (𝑅𝐴)))𝐶 ↔ ((𝐵𝐴𝐶 ∈ ran (𝑅𝐴)) ∧ 𝐵𝑅𝐶))
7 df-3an 1109 . . 3 ((𝐵𝐴𝐶 ∈ ran (𝑅𝐴) ∧ 𝐵𝑅𝐶) ↔ ((𝐵𝐴𝐶 ∈ ran (𝑅𝐴)) ∧ 𝐵𝑅𝐶))
8 3anan12 1117 . . 3 ((𝐵𝐴𝐶 ∈ ran (𝑅𝐴) ∧ 𝐵𝑅𝐶) ↔ (𝐶 ∈ ran (𝑅𝐴) ∧ (𝐵𝐴𝐵𝑅𝐶)))
96, 7, 83bitr2i 290 . 2 (𝐵(𝑅 ∩ (𝐴 × ran (𝑅𝐴)))𝐶 ↔ (𝐶 ∈ ran (𝑅𝐴) ∧ (𝐵𝐴𝐵𝑅𝐶)))
102, 5, 93bitr4i 294 1 (𝐵(𝑅𝐴)𝐶𝐵(𝑅 ∩ (𝐴 × ran (𝑅𝐴)))𝐶)
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384  w3a 1107  wcel 2155  cin 3731   class class class wbr 4809   × cxp 5275  ran crn 5278  cres 5279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-br 4810  df-opab 4872  df-xp 5283  df-rel 5284  df-cnv 5285  df-dm 5287  df-rn 5288  df-res 5289
This theorem is referenced by:  brinxprnres  34491
  Copyright terms: Public domain W3C validator