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Theorem brinxp2 4435
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 ↔ (𝐴𝐶𝐵𝐷𝐴𝑅𝐵))

Proof of Theorem brinxp2
StepHypRef Expression
1 brin 3839 . 2 (𝐴(𝑅 ∩ (𝐶 × 𝐷))𝐵 ↔ (𝐴𝑅𝐵𝐴(𝐶 × 𝐷)𝐵))
2 ancom 257 . 2 ((𝐴𝑅𝐵𝐴(𝐶 × 𝐷)𝐵) ↔ (𝐴(𝐶 × 𝐷)𝐵𝐴𝑅𝐵))
3 brxp 4403 . . . 4 (𝐴(𝐶 × 𝐷)𝐵 ↔ (𝐴𝐶𝐵𝐷))
43anbi1i 439 . . 3 ((𝐴(𝐶 × 𝐷)𝐵𝐴𝑅𝐵) ↔ ((𝐴𝐶𝐵𝐷) ∧ 𝐴𝑅𝐵))
5 df-3an 898 . . 3 ((𝐴𝐶𝐵𝐷𝐴𝑅𝐵) ↔ ((𝐴𝐶𝐵𝐷) ∧ 𝐴𝑅𝐵))
64, 5bitr4i 180 . 2 ((𝐴(𝐶 × 𝐷)𝐵𝐴𝑅𝐵) ↔ (𝐴𝐶𝐵𝐷𝐴𝑅𝐵))
71, 2, 63bitri 199 1 (𝐴(𝑅 ∩ (𝐶 × 𝐷))𝐵 ↔ (𝐴𝐶𝐵𝐷𝐴𝑅𝐵))
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102  w3a 896  wcel 1409  cin 2944   class class class wbr 3792   × cxp 4371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-xp 4379
This theorem is referenced by:  brinxp  4436  fncnv  4993  erinxp  6211
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