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Theorem elrnust 22075
Description: First direction for ustbas 22078. (Contributed by Thierry Arnoux, 16-Nov-2017.)
Assertion
Ref Expression
elrnust (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ran UnifOn)

Proof of Theorem elrnust
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elfvdm 6258 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ dom UnifOn)
2 fveq2 6229 . . . . 5 (𝑥 = 𝑋 → (UnifOn‘𝑥) = (UnifOn‘𝑋))
32eleq2d 2716 . . . 4 (𝑥 = 𝑋 → (𝑈 ∈ (UnifOn‘𝑥) ↔ 𝑈 ∈ (UnifOn‘𝑋)))
43rspcev 3340 . . 3 ((𝑋 ∈ dom UnifOn ∧ 𝑈 ∈ (UnifOn‘𝑋)) → ∃𝑥 ∈ dom UnifOn𝑈 ∈ (UnifOn‘𝑥))
51, 4mpancom 704 . 2 (𝑈 ∈ (UnifOn‘𝑋) → ∃𝑥 ∈ dom UnifOn𝑈 ∈ (UnifOn‘𝑥))
6 ustfn 22052 . . 3 UnifOn Fn V
7 fnfun 6026 . . 3 (UnifOn Fn V → Fun UnifOn)
8 elunirn 6549 . . 3 (Fun UnifOn → (𝑈 ran UnifOn ↔ ∃𝑥 ∈ dom UnifOn𝑈 ∈ (UnifOn‘𝑥)))
96, 7, 8mp2b 10 . 2 (𝑈 ran UnifOn ↔ ∃𝑥 ∈ dom UnifOn𝑈 ∈ (UnifOn‘𝑥))
105, 9sylibr 224 1 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ran UnifOn)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1523  wcel 2030  wrex 2942  Vcvv 3231   cuni 4468  dom cdm 5143  ran crn 5144  Fun wfun 5920   Fn wfn 5921  cfv 5926  UnifOncust 22050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934  df-ust 22051
This theorem is referenced by:  ustbas  22078  utopval  22083  tusval  22117  ucnval  22128  iscfilu  22139
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