Theorem brcnvin 38973

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 5162	. 2 ⊢ (𝐴(𝑅 ∩ ◡𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ 𝐴◡𝑆𝐵))
2		brcnvg 5865	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴◡𝑆𝐵 ↔ 𝐵𝑆𝐴))
3	2	anbi2d 641	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴𝑅𝐵 ∧ 𝐴◡𝑆𝐵) ↔ (𝐴𝑅𝐵 ∧ 𝐵𝑆𝐴)))
4	1, 3	bitrid 286	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝑅 ∩ ◡𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ 𝐵𝑆𝐴)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∈ wcel 2141   ∩ cin 3903   class class class wbr 5108  ^◡ccnv 5660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5256  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-cnv 5669

This theorem is referenced by:  dfantisymrel5  39460