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Theorem fneuni 33752
 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 33750 . . 3 (𝐴Fne𝐵𝐴 ⊆ (topGen‘𝐵))
21sselda 3953 . 2 ((𝐴Fne𝐵𝑆𝐴) → 𝑆 ∈ (topGen‘𝐵))
3 elfvdm 6693 . . . 4 (𝑆 ∈ (topGen‘𝐵) → 𝐵 ∈ dom topGen)
4 eltg3 21570 . . . 4 (𝐵 ∈ dom topGen → (𝑆 ∈ (topGen‘𝐵) ↔ ∃𝑥(𝑥𝐵𝑆 = 𝑥)))
53, 4syl 17 . . 3 (𝑆 ∈ (topGen‘𝐵) → (𝑆 ∈ (topGen‘𝐵) ↔ ∃𝑥(𝑥𝐵𝑆 = 𝑥)))
65ibi 270 . 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 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2115   ⊆ wss 3919  ∪ cuni 4824   class class class wbr 5052  dom cdm 5542  ‘cfv 6343  topGenctg 16711  Fnecfne 33741 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-iota 6302  df-fun 6345  df-fv 6351  df-topgen 16717  df-fne 33742 This theorem is referenced by: (None)
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