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Mirrors > Home > MPE Home > Th. List > Mathboxes > fnetg | Structured version Visualization version GIF version |
Description: A finer cover generates a topology finer than the original set. (Contributed by Mario Carneiro, 11-Sep-2015.) |
Ref | Expression |
---|---|
fnetg | ⊢ (𝐴Fne𝐵 → 𝐴 ⊆ (topGen‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . 3 ⊢ ∪ 𝐴 = ∪ 𝐴 | |
2 | eqid 2818 | . . 3 ⊢ ∪ 𝐵 = ∪ 𝐵 | |
3 | 1, 2 | isfne4 33585 | . 2 ⊢ (𝐴Fne𝐵 ↔ (∪ 𝐴 = ∪ 𝐵 ∧ 𝐴 ⊆ (topGen‘𝐵))) |
4 | 3 | simprbi 497 | 1 ⊢ (𝐴Fne𝐵 → 𝐴 ⊆ (topGen‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ⊆ wss 3933 ∪ cuni 4830 class class class wbr 5057 ‘cfv 6348 topGenctg 16699 Fnecfne 33581 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 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 16705 df-fne 33582 |
This theorem is referenced by: fnessex 33591 fneuni 33592 fnemeet2 33612 fnejoin2 33614 |
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