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Theorem br1cnvres 36928
Description: Binary relation on the converse of a restriction. (Contributed by Peter Mazsa, 27-Jul-2019.)
Assertion
Ref Expression
br1cnvres (𝐵𝑉 → (𝐵(𝑅𝐴)𝐶 ↔ (𝐶𝐴𝐶𝑅𝐵)))

Proof of Theorem br1cnvres
StepHypRef Expression
1 df-res 5680 . . . 4 (𝑅𝐴) = (𝑅 ∩ (𝐴 × V))
21cnveqi 5865 . . 3 (𝑅𝐴) = (𝑅 ∩ (𝐴 × V))
32breqi 5146 . 2 (𝐵(𝑅𝐴)𝐶𝐵(𝑅 ∩ (𝐴 × V))𝐶)
4 elex 3490 . . 3 (𝐵𝑉𝐵 ∈ V)
5 br1cnvinxp 36915 . . . . 5 (𝐵(𝑅 ∩ (𝐴 × V))𝐶 ↔ ((𝐵 ∈ V ∧ 𝐶𝐴) ∧ 𝐶𝑅𝐵))
6 anass 469 . . . . 5 (((𝐵 ∈ V ∧ 𝐶𝐴) ∧ 𝐶𝑅𝐵) ↔ (𝐵 ∈ V ∧ (𝐶𝐴𝐶𝑅𝐵)))
75, 6bitri 274 . . . 4 (𝐵(𝑅 ∩ (𝐴 × V))𝐶 ↔ (𝐵 ∈ V ∧ (𝐶𝐴𝐶𝑅𝐵)))
87baib 536 . . 3 (𝐵 ∈ V → (𝐵(𝑅 ∩ (𝐴 × V))𝐶 ↔ (𝐶𝐴𝐶𝑅𝐵)))
94, 8syl 17 . 2 (𝐵𝑉 → (𝐵(𝑅 ∩ (𝐴 × V))𝐶 ↔ (𝐶𝐴𝐶𝑅𝐵)))
103, 9bitrid 282 1 (𝐵𝑉 → (𝐵(𝑅𝐴)𝐶 ↔ (𝐶𝐴𝐶𝑅𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wcel 2106  Vcvv 3472  cin 3942   class class class wbr 5140   × cxp 5666  ccnv 5667  cres 5670
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5291  ax-nul 5298  ax-pr 5419
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3474  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5141  df-opab 5203  df-xp 5674  df-rel 5675  df-cnv 5676  df-res 5680
This theorem is referenced by:  coss1cnvres  37078
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