Theorem br1cnvinxp 37614

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 5804	. . 3 ⊢ Rel (𝑅 ∩ (𝐴 × 𝐵))
2	1	relbrcnv 6096	. 2 ⊢ (𝐶◡(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ 𝐷(𝑅 ∩ (𝐴 × 𝐵))𝐶)
3		brinxp2 5743	. 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 205   ∧ wa 395   ∈ wcel 2098   ∩ cin 3939   class class class wbr 5138   × cxp 5664  ^◡ccnv 5665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-xp 5672  df-rel 5673  df-cnv 5674

This theorem is referenced by:  br1cnvres  37627