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Theorem ustrel 21925
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 21918 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → 𝑉 ⊆ (𝑋 × 𝑋))
2 xpss 5187 . . 3 (𝑋 × 𝑋) ⊆ (V × V)
31, 2syl6ss 3595 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → 𝑉 ⊆ (V × V))
4 df-rel 5081 . 2 (Rel 𝑉𝑉 ⊆ (V × V))
53, 4sylibr 224 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → Rel 𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  Vcvv 3186  wss 3555   × cxp 5072  Rel wrel 5079  cfv 5847  UnifOncust 21913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-res 5086  df-iota 5810  df-fun 5849  df-fv 5855  df-ust 21914
This theorem is referenced by:  ustssco  21928  ustexsym  21929  ustuqtop4  21958  utop2nei  21964  utop3cls  21965  ucncn  21999
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