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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 21583 | . . . 4 ⊢ (𝐾 ∈ Top → (topGen‘𝐾) = 𝐾) | |
2 | 1 | sseq2d 4001 | . . 3 ⊢ (𝐾 ∈ Top → (𝐽 ⊆ (topGen‘𝐾) ↔ 𝐽 ⊆ 𝐾)) |
3 | 2 | bicomd 225 | . 2 ⊢ (𝐾 ∈ Top → (𝐽 ⊆ 𝐾 ↔ 𝐽 ⊆ (topGen‘𝐾))) |
4 | topfne.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
5 | topfne.2 | . . . 4 ⊢ 𝑌 = ∪ 𝐾 | |
6 | 4, 5 | isfne4 33690 | . . 3 ⊢ (𝐽Fne𝐾 ↔ (𝑋 = 𝑌 ∧ 𝐽 ⊆ (topGen‘𝐾))) |
7 | 6 | baibr 539 | . 2 ⊢ (𝑋 = 𝑌 → (𝐽 ⊆ (topGen‘𝐾) ↔ 𝐽Fne𝐾)) |
8 | 3, 7 | sylan9bb 512 | 1 ⊢ ((𝐾 ∈ Top ∧ 𝑋 = 𝑌) → (𝐽 ⊆ 𝐾 ↔ 𝐽Fne𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 ∪ cuni 4840 class class class wbr 5068 ‘cfv 6357 topGenctg 16713 Topctop 21503 Fnecfne 33686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-topgen 16719 df-top 21504 df-fne 33687 |
This theorem is referenced by: (None) |
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