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Theorem isufl 22509
Description: Define the (strong) ultrafilter lemma, parameterized over base sets. A set 𝑋 satisfies the ultrafilter lemma if every filter on 𝑋 is a subset of some ultrafilter. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
isufl (𝑋𝑉 → (𝑋 ∈ UFL ↔ ∀𝑓 ∈ (Fil‘𝑋)∃𝑔 ∈ (UFil‘𝑋)𝑓𝑔))
Distinct variable group:   𝑓,𝑔,𝑋
Allowed substitution hints:   𝑉(𝑓,𝑔)

Proof of Theorem isufl
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6653 . . 3 (𝑥 = 𝑋 → (Fil‘𝑥) = (Fil‘𝑋))
2 fveq2 6653 . . . 4 (𝑥 = 𝑋 → (UFil‘𝑥) = (UFil‘𝑋))
32rexeqdv 3403 . . 3 (𝑥 = 𝑋 → (∃𝑔 ∈ (UFil‘𝑥)𝑓𝑔 ↔ ∃𝑔 ∈ (UFil‘𝑋)𝑓𝑔))
41, 3raleqbidv 3392 . 2 (𝑥 = 𝑋 → (∀𝑓 ∈ (Fil‘𝑥)∃𝑔 ∈ (UFil‘𝑥)𝑓𝑔 ↔ ∀𝑓 ∈ (Fil‘𝑋)∃𝑔 ∈ (UFil‘𝑋)𝑓𝑔))
5 df-ufl 22498 . 2 UFL = {𝑥 ∣ ∀𝑓 ∈ (Fil‘𝑥)∃𝑔 ∈ (UFil‘𝑥)𝑓𝑔}
64, 5elab2g 3653 1 (𝑋𝑉 → (𝑋 ∈ UFL ↔ ∀𝑓 ∈ (Fil‘𝑋)∃𝑔 ∈ (UFil‘𝑋)𝑓𝑔))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wcel 2115  wral 3132  wrex 3133  wss 3918  cfv 6338  Filcfil 22441  UFilcufil 22495  UFLcufl 22496
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
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-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4822  df-br 5050  df-iota 6297  df-fv 6346  df-ufl 22498
This theorem is referenced by:  ufli  22510  numufl  22511  ssufl  22514  ufldom  22558
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