Theorem br1cnvinxp 38212

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 ↔ ((𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) ∧ 𝐷𝑅𝐶))

Proof of Theorem br1cnvinxp

Step	Hyp	Ref	Expression
1		relinxp 5838	. . 3 ⊢ Rel (𝑅 ∩ (𝐴 × 𝐵))
2	1	relbrcnv 6137	. 2 ⊢ (𝐶◡(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ 𝐷(𝑅 ∩ (𝐴 × 𝐵))𝐶)
3		brinxp2 5777	. 2 ⊢ (𝐷(𝑅 ∩ (𝐴 × 𝐵))𝐶 ↔ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) ∧ 𝐷𝑅𝐶))
4		ancom 460	. . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) ↔ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴))
5	4	anbi1i 623	. 2 ⊢ (((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) ∧ 𝐷𝑅𝐶) ↔ ((𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) ∧ 𝐷𝑅𝐶))
6	2, 3, 5	3bitri 297	1 ⊢ (𝐶◡(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ ((𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) ∧ 𝐷𝑅𝐶))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2108   ∩ cin 3975   class class class wbr 5166   × cxp 5698  ^◡ccnv 5699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-cnv 5708

This theorem is referenced by:  br1cnvres  38225