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Theorem ustrel 22812
Description: The elements of uniform structures, called entourages, are relations. (Contributed by Thierry Arnoux, 15-Nov-2017.)
Assertion
Ref Expression
ustrel ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → Rel 𝑉)

Proof of Theorem ustrel
StepHypRef Expression
1 ustssxp 22805 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → 𝑉 ⊆ (𝑋 × 𝑋))
2 xpss 5564 . . 3 (𝑋 × 𝑋) ⊆ (V × V)
31, 2sstrdi 3977 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → 𝑉 ⊆ (V × V))
4 df-rel 5555 . 2 (Rel 𝑉𝑉 ⊆ (V × V))
53, 4sylibr 236 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → Rel 𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wcel 2107  Vcvv 3493  wss 3934   × cxp 5546  Rel wrel 5553  cfv 6348  UnifOncust 22800
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-res 5560  df-iota 6307  df-fun 6350  df-fv 6356  df-ust 22801
This theorem is referenced by:  ustssco  22815  ustexsym  22816  ustuqtop4  22845  utop2nei  22851  utop3cls  22852  ucncn  22886
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