Theorem brinxp2 5721

Description: Intersection of binary relation with Cartesian product. (Contributed by NM, 3-Mar-2007.) (Revised by Mario Carneiro, 26-Apr-2015.) Group conjuncts and avoid df-3an 1099. (Revised by Peter Mazsa, 18-Sep-2022.)

Assertion

Ref	Expression
brinxp2	⊢ (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 𝐶𝑅𝐷))

Proof of Theorem brinxp2

Step	Hyp	Ref	Expression
1		brin 5149	. 2 ⊢ (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ (𝐶𝑅𝐷 ∧ 𝐶(𝐴 × 𝐵)𝐷))
2		ancom 464	. 2 ⊢ ((𝐶𝑅𝐷 ∧ 𝐶(𝐴 × 𝐵)𝐷) ↔ (𝐶(𝐴 × 𝐵)𝐷 ∧ 𝐶𝑅𝐷))
3		brxp 5692	. . 3 ⊢ (𝐶(𝐴 × 𝐵)𝐷 ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵))
4	3	anbi1i 633	. 2 ⊢ ((𝐶(𝐴 × 𝐵)𝐷 ∧ 𝐶𝑅𝐷) ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 𝐶𝑅𝐷))
5	1, 2, 4	3bitri 299	1 ⊢ (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 𝐶𝑅𝐷))

Syntax hints:   ↔ wb 208   ∧ wa 399   ∈ wcel 2141   ∩ cin 3901   class class class wbr 5097   × cxp 5641

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5243  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-xp 5649

This theorem is referenced by:  brinxp  5722  opelinxp  5723  fncnv  6589  erinxp  8767  fpwwe2lem7  10589  fpwwe2lem8  10590  fpwwe2lem11  10593  nqerf  10882  nqerid  10885  isstruct  17179  pwsle  17513  psss  18603  psssdm2  18604  pi1cpbl  25094  pi1grplem  25099  br1cnvinxp  38719  brres2  38733  inxpss  38777  inxpss3  38780  idinxpssinxp2  38784  inxp2  38835  inxpxrn  38878