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Theorem elrnust 22748
Description: First direction for ustbas 22751. (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 6699 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ dom UnifOn)
2 fveq2 6667 . . . . 5 (𝑥 = 𝑋 → (UnifOn‘𝑥) = (UnifOn‘𝑋))
32eleq2d 2903 . . . 4 (𝑥 = 𝑋 → (𝑈 ∈ (UnifOn‘𝑥) ↔ 𝑈 ∈ (UnifOn‘𝑋)))
43rspcev 3627 . . 3 ((𝑋 ∈ dom UnifOn ∧ 𝑈 ∈ (UnifOn‘𝑋)) → ∃𝑥 ∈ dom UnifOn𝑈 ∈ (UnifOn‘𝑥))
51, 4mpancom 684 . 2 (𝑈 ∈ (UnifOn‘𝑋) → ∃𝑥 ∈ dom UnifOn𝑈 ∈ (UnifOn‘𝑥))
6 ustfn 22725 . . 3 UnifOn Fn V
7 fnfun 6450 . . 3 (UnifOn Fn V → Fun UnifOn)
8 elunirn 7004 . . 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 235 1 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ran UnifOn)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1530  wcel 2107  wrex 3144  Vcvv 3500   cuni 4837  dom cdm 5554  ran crn 5555  Fun wfun 6346   Fn wfn 6347  cfv 6352  UnifOncust 22723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-iota 6312  df-fun 6354  df-fn 6355  df-fv 6360  df-ust 22724
This theorem is referenced by:  ustbas  22751  utopval  22756  tusval  22790  ucnval  22801  iscfilu  22812
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