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Mirrors > Home > MPE Home > Th. List > ustrel | Structured version Visualization version GIF version |
Description: The elements of uniform structures, called entourages, are relations. (Contributed by Thierry Arnoux, 15-Nov-2017.) |
Ref | Expression |
---|---|
ustrel | ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → Rel 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ustssxp 22810 | . . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → 𝑉 ⊆ (𝑋 × 𝑋)) | |
2 | xpss 5535 | . . 3 ⊢ (𝑋 × 𝑋) ⊆ (V × V) | |
3 | 1, 2 | sstrdi 3927 | . 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → 𝑉 ⊆ (V × V)) |
4 | df-rel 5526 | . 2 ⊢ (Rel 𝑉 ↔ 𝑉 ⊆ (V × V)) | |
5 | 3, 4 | sylibr 237 | 1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → Rel 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 × cxp 5517 Rel wrel 5524 ‘cfv 6324 UnifOncust 22805 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-res 5531 df-iota 6283 df-fun 6326 df-fv 6332 df-ust 22806 |
This theorem is referenced by: ustssco 22820 ustexsym 22821 ustuqtop4 22850 utop2nei 22856 utop3cls 22857 ucncn 22891 |
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