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Theorem brres2 36757
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 5949 . . 3 (𝐶 ∈ ran (𝑅𝐴) → (𝐵(𝑅𝐴)𝐶 ↔ (𝐵𝐴𝐵𝑅𝐶)))
21pm5.32i 576 . 2 ((𝐶 ∈ ran (𝑅𝐴) ∧ 𝐵(𝑅𝐴)𝐶) ↔ (𝐶 ∈ ran (𝑅𝐴) ∧ (𝐵𝐴𝐵𝑅𝐶)))
3 relres 5971 . . . 4 Rel (𝑅𝐴)
43relelrni 5909 . . 3 (𝐵(𝑅𝐴)𝐶𝐶 ∈ ran (𝑅𝐴))
54pm4.71ri 562 . 2 (𝐵(𝑅𝐴)𝐶 ↔ (𝐶 ∈ ran (𝑅𝐴) ∧ 𝐵(𝑅𝐴)𝐶))
6 brinxp2 5714 . . 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 3914   class class class wbr 5110   × cxp 5636  ran crn 5639  cres 5640
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 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
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 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-xp 5644  df-rel 5645  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650
This theorem is referenced by:  brinxprnres  36781
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