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Theorem inxpssres 34412
 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 3771 . . . 4 𝐴𝐴
2 ssv 3772 . . . 4 𝐵 ⊆ V
3 xpss12 5264 . . . 4 ((𝐴𝐴𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (𝐴 × V))
41, 2, 3mp2an 664 . . 3 (𝐴 × 𝐵) ⊆ (𝐴 × V)
5 sslin 3985 . . 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 5261 . 2 (𝑅𝐴) = (𝑅 ∩ (𝐴 × V))
86, 7sseqtr4i 3785 1 (𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝑅𝐴)
 Colors of variables: wff setvar class Syntax hints:  Vcvv 3349   ∩ cin 3720   ⊆ wss 3721   × cxp 5247   ↾ cres 5251 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-v 3351  df-in 3728  df-ss 3735  df-opab 4845  df-xp 5255  df-res 5261 This theorem is referenced by:  idreseqidinxp  34416  idinxpres  34424
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