Theorem brcnvin 37543

Description: Intersection with a converse, binary relation. (Contributed by Peter Mazsa, 24-Mar-2024.)

Assertion

Ref	Expression
brcnvin	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝑅 ∩ ◡𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ 𝐵𝑆𝐴)))

Proof of Theorem brcnvin

Step	Hyp	Ref	Expression
1		brin 5200	. 2 ⊢ (𝐴(𝑅 ∩ ◡𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ 𝐴◡𝑆𝐵))
2		brcnvg 5879	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴◡𝑆𝐵 ↔ 𝐵𝑆𝐴))
3	2	anbi2d 629	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴𝑅𝐵 ∧ 𝐴◡𝑆𝐵) ↔ (𝐴𝑅𝐵 ∧ 𝐵𝑆𝐴)))
4	1, 3	bitrid 282	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝑅 ∩ ◡𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ 𝐵𝑆𝐴)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∈ wcel 2106   ∩ cin 3947   class class class wbr 5148  ^◡ccnv 5675

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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

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 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-cnv 5684

This theorem is referenced by:  dfantisymrel5  37935