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Mirrors > Home > MPE Home > Th. List > Mathboxes > topfne | Structured version Visualization version GIF version |
Description: Fineness for covers corresponds precisely with fineness for topologies. (Contributed by Jeff Hankins, 29-Sep-2009.) |
Ref | Expression |
---|---|
topfne.1 | ⊢ 𝑋 = ∪ 𝐽 |
topfne.2 | ⊢ 𝑌 = ∪ 𝐾 |
Ref | Expression |
---|---|
topfne | ⊢ ((𝐾 ∈ Top ∧ 𝑋 = 𝑌) → (𝐽 ⊆ 𝐾 ↔ 𝐽Fne𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgtop 20588 | . . . 4 ⊢ (𝐾 ∈ Top → (topGen‘𝐾) = 𝐾) | |
2 | 1 | sseq2d 3596 | . . 3 ⊢ (𝐾 ∈ Top → (𝐽 ⊆ (topGen‘𝐾) ↔ 𝐽 ⊆ 𝐾)) |
3 | 2 | bicomd 212 | . 2 ⊢ (𝐾 ∈ Top → (𝐽 ⊆ 𝐾 ↔ 𝐽 ⊆ (topGen‘𝐾))) |
4 | topfne.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
5 | topfne.2 | . . . 4 ⊢ 𝑌 = ∪ 𝐾 | |
6 | 4, 5 | isfne4 31505 | . . 3 ⊢ (𝐽Fne𝐾 ↔ (𝑋 = 𝑌 ∧ 𝐽 ⊆ (topGen‘𝐾))) |
7 | 6 | baibr 943 | . 2 ⊢ (𝑋 = 𝑌 → (𝐽 ⊆ (topGen‘𝐾) ↔ 𝐽Fne𝐾)) |
8 | 3, 7 | sylan9bb 732 | 1 ⊢ ((𝐾 ∈ Top ∧ 𝑋 = 𝑌) → (𝐽 ⊆ 𝐾 ↔ 𝐽Fne𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 ∪ cuni 4372 class class class wbr 4583 ‘cfv 5804 topGenctg 15921 Topctop 20517 Fnecfne 31501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-topgen 15927 df-top 20521 df-fne 31502 |
This theorem is referenced by: (None) |
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