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Mirrors > Home > MPE Home > Th. List > Mathboxes > fneuni | Structured version Visualization version GIF version |
Description: If 𝐵 is finer than 𝐴, every element of 𝐴 is a union of elements of 𝐵. (Contributed by Jeff Hankins, 11-Oct-2009.) |
Ref | Expression |
---|---|
fneuni | ⊢ ((𝐴Fne𝐵 ∧ 𝑆 ∈ 𝐴) → ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝑆 = ∪ 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnetg 32671 | . . 3 ⊢ (𝐴Fne𝐵 → 𝐴 ⊆ (topGen‘𝐵)) | |
2 | 1 | sselda 3750 | . 2 ⊢ ((𝐴Fne𝐵 ∧ 𝑆 ∈ 𝐴) → 𝑆 ∈ (topGen‘𝐵)) |
3 | elfvdm 6361 | . . . 4 ⊢ (𝑆 ∈ (topGen‘𝐵) → 𝐵 ∈ dom topGen) | |
4 | eltg3 20986 | . . . 4 ⊢ (𝐵 ∈ dom topGen → (𝑆 ∈ (topGen‘𝐵) ↔ ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝑆 = ∪ 𝑥))) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝑆 ∈ (topGen‘𝐵) → (𝑆 ∈ (topGen‘𝐵) ↔ ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝑆 = ∪ 𝑥))) |
6 | 5 | ibi 256 | . 2 ⊢ (𝑆 ∈ (topGen‘𝐵) → ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝑆 = ∪ 𝑥)) |
7 | 2, 6 | syl 17 | 1 ⊢ ((𝐴Fne𝐵 ∧ 𝑆 ∈ 𝐴) → ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝑆 = ∪ 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1630 ∃wex 1851 ∈ wcel 2144 ⊆ wss 3721 ∪ cuni 4572 class class class wbr 4784 dom cdm 5249 ‘cfv 6031 topGenctg 16305 Fnecfne 32662 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-8 2146 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-sep 4912 ax-nul 4920 ax-pow 4971 ax-pr 5034 ax-un 7095 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3an 1072 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ne 2943 df-ral 3065 df-rex 3066 df-rab 3069 df-v 3351 df-sbc 3586 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-nul 4062 df-if 4224 df-pw 4297 df-sn 4315 df-pr 4317 df-op 4321 df-uni 4573 df-br 4785 df-opab 4845 df-mpt 4862 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-iota 5994 df-fun 6033 df-fv 6039 df-topgen 16311 df-fne 32663 |
This theorem is referenced by: (None) |
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