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Theorem isufl 21469
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 6088 . . 3 (𝑥 = 𝑋 → (Fil‘𝑥) = (Fil‘𝑋))
2 fveq2 6088 . . . 4 (𝑥 = 𝑋 → (UFil‘𝑥) = (UFil‘𝑋))
32rexeqdv 3121 . . 3 (𝑥 = 𝑋 → (∃𝑔 ∈ (UFil‘𝑥)𝑓𝑔 ↔ ∃𝑔 ∈ (UFil‘𝑋)𝑓𝑔))
41, 3raleqbidv 3128 . 2 (𝑥 = 𝑋 → (∀𝑓 ∈ (Fil‘𝑥)∃𝑔 ∈ (UFil‘𝑥)𝑓𝑔 ↔ ∀𝑓 ∈ (Fil‘𝑋)∃𝑔 ∈ (UFil‘𝑋)𝑓𝑔))
5 df-ufl 21458 . 2 UFL = {𝑥 ∣ ∀𝑓 ∈ (Fil‘𝑥)∃𝑔 ∈ (UFil‘𝑥)𝑓𝑔}
64, 5elab2g 3321 1 (𝑋𝑉 → (𝑋 ∈ UFL ↔ ∀𝑓 ∈ (Fil‘𝑋)∃𝑔 ∈ (UFil‘𝑋)𝑓𝑔))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194   = wceq 1474  wcel 1976  wral 2895  wrex 2896  wss 3539  cfv 5790  Filcfil 21401  UFilcufil 21455  UFLcufl 21456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-iota 5754  df-fv 5798  df-ufl 21458
This theorem is referenced by:  ufli  21470  numufl  21471  ssufl  21474  ufldom  21518
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