Theorem trinxp 6123

Description: The relation induced by a transitive relation on a part of its field is transitive. (Taking the intersection of a relation with a Cartesian square is a way to restrict it to a subset of its field.) (Contributed by FL, 31-Jul-2009.)

Assertion

Ref	Expression
trinxp	⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 → ((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ (𝑅 ∩ (𝐴 × 𝐴)))

Proof of Theorem trinxp

Step	Hyp	Ref	Expression
1		xpidtr 6120	. 2 ⊢ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)
2		trin2 6121	. 2 ⊢ (((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)) → ((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ (𝑅 ∩ (𝐴 × 𝐴)))
3	1, 2	mpan2 704	1 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 → ((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ (𝑅 ∩ (𝐴 × 𝐴)))

This proof depends on syntax axioms:   → wi 4   ∩ cin 3901   ⊆ wss 3902   × cxp 5657   ∘ ccom 5663

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-xp 5665  df-rel 5666  df-co 5668

This theorem is used by:  psss  18674