Theorem relinxp 5797

Description: Intersection with a Cartesian product is a relation. (Contributed by Peter Mazsa, 4-Mar-2019.)

Assertion

Ref	Expression
relinxp	⊢ Rel (𝑅 ∩ (𝐴 × 𝐵))

Proof of Theorem relinxp

Step	Hyp	Ref	Expression
1		relxp 5673	. 2 ⊢ Rel (𝐴 × 𝐵)
2		relin2 5796	. 2 ⊢ (Rel (𝐴 × 𝐵) → Rel (𝑅 ∩ (𝐴 × 𝐵)))
3	1, 2	ax-mp 5	1 ⊢ Rel (𝑅 ∩ (𝐴 × 𝐵))

This proof depends on syntax axioms:   ∩ cin 4092   × cxp 5653  Rel wrel 5660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-in 4094  df-opab 5171  df-xp 5661  df-rel 5662

This theorem is used by:  inxp  5815  relres  6007  elinxp  6020  cnvcnv  6196  tpostpos  8291  brinxper  8775  erinxp  8840  brdom3  10634  brdom5  10635  brdom4  10636  fpwwe2lem7  10749  fpwwe2lem8  10750  fpwwe2lem11  10753  pwsle  17693  opsrtoslem2  22395  cgraer  29407  elrn3  36705  bj-idres  38273  br1cnvinxp  39383  inxprnres  39422  inxpss  39441  inxpss2  39445  iss2  39468  inxp2  39499  inxpxrn  39542