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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 31510 | . . 3 ⊢ (𝐴Fne𝐵 → 𝐴 ⊆ (topGen‘𝐵)) | |
2 | 1 | sselda 3568 | . 2 ⊢ ((𝐴Fne𝐵 ∧ 𝑆 ∈ 𝐴) → 𝑆 ∈ (topGen‘𝐵)) |
3 | elfvdm 6130 | . . . 4 ⊢ (𝑆 ∈ (topGen‘𝐵) → 𝐵 ∈ dom topGen) | |
4 | eltg3 20577 | . . . 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 255 | . 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 195 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ⊆ wss 3540 ∪ cuni 4372 class class class wbr 4583 dom cdm 5038 ‘cfv 5804 topGenctg 15921 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-fne 31502 |
This theorem is referenced by: (None) |
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