Mathbox for Peter Mazsa < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  inxpssres Structured version   Visualization version   GIF version

Theorem inxpssres 34400
 Description: Intersection with a Cartesian product is a subclass of restriction. (Contributed by Peter Mazsa, 19-Jul-2019.)
Assertion
Ref Expression
inxpssres (𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝑅𝐴)

Proof of Theorem inxpssres
StepHypRef Expression
1 ssid 3765 . . . 4 𝐴𝐴
2 ssv 3766 . . . 4 𝐵 ⊆ V
3 xpss12 5281 . . . 4 ((𝐴𝐴𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (𝐴 × V))
41, 2, 3mp2an 710 . . 3 (𝐴 × 𝐵) ⊆ (𝐴 × V)
5 sslin 3982 . . 3 ((𝐴 × 𝐵) ⊆ (𝐴 × V) → (𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝑅 ∩ (𝐴 × V)))
64, 5ax-mp 5 . 2 (𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝑅 ∩ (𝐴 × V))
7 df-res 5278 . 2 (𝑅𝐴) = (𝑅 ∩ (𝐴 × V))
86, 7sseqtr4i 3779 1 (𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝑅𝐴)
 Colors of variables: wff setvar class Syntax hints:  Vcvv 3340   ∩ cin 3714   ⊆ wss 3715   × cxp 5264   ↾ cres 5268 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-in 3722  df-ss 3729  df-opab 4865  df-xp 5272  df-res 5278 This theorem is referenced by:  idreseqidinxp  34404  idinxpres  34412
 Copyright terms: Public domain W3C validator