Theorem br1cnvinxp 38908

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 5801	. . 3 ⊢ Rel (𝑅 ∩ (𝐴 × 𝐵))
2	1	relbrcnv 6109	. 2 ⊢ (𝐶◡(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ 𝐷(𝑅 ∩ (𝐴 × 𝐵))𝐶)
3		brinxp2 5739	. 2 ⊢ (𝐷(𝑅 ∩ (𝐴 × 𝐵))𝐶 ↔ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) ∧ 𝐷𝑅𝐶))
4		ancom 465	. . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) ↔ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴))
5	4	anbi1i 635	. 2 ⊢ (((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) ∧ 𝐷𝑅𝐶) ↔ ((𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) ∧ 𝐷𝑅𝐶))
6	2, 3, 5	3bitri 300	1 ⊢ (𝐶◡(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ ((𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) ∧ 𝐷𝑅𝐶))

Syntax hints:   ↔ wb 209   ∧ wa 400   ∈ wcel 2143   ∩ cin 3904   class class class wbr 5109   × cxp 5659  ^◡ccnv 5660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-xp 5667  df-rel 5668  df-cnv 5669

This theorem is referenced by:  br1cnvres  38923