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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 21573 | . . . 4 ⊢ (𝐾 ∈ Top → (topGen‘𝐾) = 𝐾) | |
2 | 1 | sseq2d 3997 | . . 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 33681 | . . 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 1531 ∈ wcel 2108 ⊆ wss 3934 ∪ cuni 4830 class class class wbr 5057 ‘cfv 6348 topGenctg 16703 Topctop 21493 Fnecfne 33677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 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 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 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-iota 6307 df-fun 6350 df-fv 6356 df-topgen 16709 df-top 21494 df-fne 33678 |
This theorem is referenced by: (None) |
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