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Theorem brres2 37042
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 5983 . . 3 (𝐶 ∈ ran (𝑅𝐴) → (𝐵(𝑅𝐴)𝐶 ↔ (𝐵𝐴𝐵𝑅𝐶)))
21pm5.32i 576 . 2 ((𝐶 ∈ ran (𝑅𝐴) ∧ 𝐵(𝑅𝐴)𝐶) ↔ (𝐶 ∈ ran (𝑅𝐴) ∧ (𝐵𝐴𝐵𝑅𝐶)))
3 relres 6005 . . . 4 Rel (𝑅𝐴)
43relelrni 5943 . . 3 (𝐵(𝑅𝐴)𝐶𝐶 ∈ ran (𝑅𝐴))
54pm4.71ri 562 . 2 (𝐵(𝑅𝐴)𝐶 ↔ (𝐶 ∈ ran (𝑅𝐴) ∧ 𝐵(𝑅𝐴)𝐶))
6 brinxp2 5748 . . 3 (𝐵(𝑅 ∩ (𝐴 × ran (𝑅𝐴)))𝐶 ↔ ((𝐵𝐴𝐶 ∈ ran (𝑅𝐴)) ∧ 𝐵𝑅𝐶))
7 df-3an 1090 . . 3 ((𝐵𝐴𝐶 ∈ ran (𝑅𝐴) ∧ 𝐵𝑅𝐶) ↔ ((𝐵𝐴𝐶 ∈ ran (𝑅𝐴)) ∧ 𝐵𝑅𝐶))
8 3anan12 1097 . . 3 ((𝐵𝐴𝐶 ∈ ran (𝑅𝐴) ∧ 𝐵𝑅𝐶) ↔ (𝐶 ∈ ran (𝑅𝐴) ∧ (𝐵𝐴𝐵𝑅𝐶)))
96, 7, 83bitr2i 299 . 2 (𝐵(𝑅 ∩ (𝐴 × ran (𝑅𝐴)))𝐶 ↔ (𝐶 ∈ ran (𝑅𝐴) ∧ (𝐵𝐴𝐵𝑅𝐶)))
102, 5, 93bitr4i 303 1 (𝐵(𝑅𝐴)𝐶𝐵(𝑅 ∩ (𝐴 × ran (𝑅𝐴)))𝐶)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  w3a 1088  wcel 2107  cin 3945   class class class wbr 5144   × cxp 5670  ran crn 5673  cres 5674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5145  df-opab 5207  df-xp 5678  df-rel 5679  df-cnv 5680  df-dm 5682  df-rn 5683  df-res 5684
This theorem is referenced by:  brinxprnres  37066
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