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Theorem fneuni 32673
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.)
Assertion
Ref Expression
fneuni ((𝐴Fne𝐵𝑆𝐴) → ∃𝑥(𝑥𝐵𝑆 = 𝑥))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑆

Proof of Theorem fneuni
StepHypRef Expression
1 fnetg 32671 . . 3 (𝐴Fne𝐵𝐴 ⊆ (topGen‘𝐵))
21sselda 3750 . 2 ((𝐴Fne𝐵𝑆𝐴) → 𝑆 ∈ (topGen‘𝐵))
3 elfvdm 6361 . . . 4 (𝑆 ∈ (topGen‘𝐵) → 𝐵 ∈ dom topGen)
4 eltg3 20986 . . . 4 (𝐵 ∈ dom topGen → (𝑆 ∈ (topGen‘𝐵) ↔ ∃𝑥(𝑥𝐵𝑆 = 𝑥)))
53, 4syl 17 . . 3 (𝑆 ∈ (topGen‘𝐵) → (𝑆 ∈ (topGen‘𝐵) ↔ ∃𝑥(𝑥𝐵𝑆 = 𝑥)))
65ibi 256 . 2 (𝑆 ∈ (topGen‘𝐵) → ∃𝑥(𝑥𝐵𝑆 = 𝑥))
72, 6syl 17 1 ((𝐴Fne𝐵𝑆𝐴) → ∃𝑥(𝑥𝐵𝑆 = 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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