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Theorem br1cnvinxp 38257
Description: Binary relation on the converse of an intersection with a Cartesian product. (Contributed by Peter Mazsa, 27-Jul-2019.)
Assertion
Ref Expression
br1cnvinxp (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ ((𝐶𝐵𝐷𝐴) ∧ 𝐷𝑅𝐶))

Proof of Theorem br1cnvinxp
StepHypRef Expression
1 relinxp 5824 . . 3 Rel (𝑅 ∩ (𝐴 × 𝐵))
21relbrcnv 6125 . 2 (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷𝐷(𝑅 ∩ (𝐴 × 𝐵))𝐶)
3 brinxp2 5763 . 2 (𝐷(𝑅 ∩ (𝐴 × 𝐵))𝐶 ↔ ((𝐷𝐴𝐶𝐵) ∧ 𝐷𝑅𝐶))
4 ancom 460 . . 3 ((𝐷𝐴𝐶𝐵) ↔ (𝐶𝐵𝐷𝐴))
54anbi1i 624 . 2 (((𝐷𝐴𝐶𝐵) ∧ 𝐷𝑅𝐶) ↔ ((𝐶𝐵𝐷𝐴) ∧ 𝐷𝑅𝐶))
62, 3, 53bitri 297 1 (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ ((𝐶𝐵𝐷𝐴) ∧ 𝐷𝑅𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wcel 2108  cin 3950   class class class wbr 5143   × cxp 5683  ccnv 5684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693
This theorem is referenced by:  br1cnvres  38270
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