Theorem ustrel 24067

Description: The elements of uniform structures, called entourages, are relations. (Contributed by Thierry Arnoux, 15-Nov-2017.)

Assertion

Ref	Expression
ustrel	⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → Rel 𝑉)

Proof of Theorem ustrel

Step	Hyp	Ref	Expression
1		ustssxp 24060	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → 𝑉 ⊆ (𝑋 × 𝑋))
2		xpss 5685	. . 3 ⊢ (𝑋 × 𝑋) ⊆ (V × V)
3	1, 2	sstrdi 3989	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → 𝑉 ⊆ (V × V))
4		df-rel 5676	. 2 ⊢ (Rel 𝑉 ↔ 𝑉 ⊆ (V × V))
5	3, 4	sylibr 233	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → Rel 𝑉)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2098  Vcvv 3468   ⊆ wss 3943   × cxp 5667  Rel wrel 5674  ‘cfv 6536  UnifOncust 24055

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-res 5681  df-iota 6488  df-fun 6538  df-fv 6544  df-ust 24056

This theorem is referenced by:  ustssco  24070  ustexsym  24071  ustuqtop4  24100  utop2nei  24106  utop3cls  24107  ucncn  24141