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Theorem brinxp2 4350
Description: Intersection of binary relation with cross product. (Contributed by NM, 3-Mar-2007.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
brinxp2 (A(𝑅 ∩ (𝐶 × 𝐷))B ↔ (A 𝐶 B 𝐷 A𝑅B))

Proof of Theorem brinxp2
StepHypRef Expression
1 brin 3802 . 2 (A(𝑅 ∩ (𝐶 × 𝐷))B ↔ (A𝑅B A(𝐶 × 𝐷)B))
2 ancom 253 . 2 ((A𝑅B A(𝐶 × 𝐷)B) ↔ (A(𝐶 × 𝐷)B A𝑅B))
3 brxp 4318 . . . 4 (A(𝐶 × 𝐷)B ↔ (A 𝐶 B 𝐷))
43anbi1i 431 . . 3 ((A(𝐶 × 𝐷)B A𝑅B) ↔ ((A 𝐶 B 𝐷) A𝑅B))
5 df-3an 886 . . 3 ((A 𝐶 B 𝐷 A𝑅B) ↔ ((A 𝐶 B 𝐷) A𝑅B))
64, 5bitr4i 176 . 2 ((A(𝐶 × 𝐷)B A𝑅B) ↔ (A 𝐶 B 𝐷 A𝑅B))
71, 2, 63bitri 195 1 (A(𝑅 ∩ (𝐶 × 𝐷))B ↔ (A 𝐶 B 𝐷 A𝑅B))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   w3a 884   wcel 1390  cin 2910   class class class wbr 3755   × cxp 4286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294
This theorem is referenced by:  brinxp  4351  fncnv  4908  erinxp  6116
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