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Theorem elrnust 22248
Description: First direction for ustbas 22251. (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 6363 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ dom UnifOn)
2 fveq2 6333 . . . . 5 (𝑥 = 𝑋 → (UnifOn‘𝑥) = (UnifOn‘𝑋))
32eleq2d 2836 . . . 4 (𝑥 = 𝑋 → (𝑈 ∈ (UnifOn‘𝑥) ↔ 𝑈 ∈ (UnifOn‘𝑋)))
43rspcev 3460 . . 3 ((𝑋 ∈ dom UnifOn ∧ 𝑈 ∈ (UnifOn‘𝑋)) → ∃𝑥 ∈ dom UnifOn𝑈 ∈ (UnifOn‘𝑥))
51, 4mpancom 668 . 2 (𝑈 ∈ (UnifOn‘𝑋) → ∃𝑥 ∈ dom UnifOn𝑈 ∈ (UnifOn‘𝑥))
6 ustfn 22225 . . 3 UnifOn Fn V
7 fnfun 6127 . . 3 (UnifOn Fn V → Fun UnifOn)
8 elunirn 6655 . . 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 1631  wcel 2145  wrex 3062  Vcvv 3351   cuni 4575  dom cdm 5250  ran crn 5251  Fun wfun 6024   Fn wfn 6025  cfv 6030  UnifOncust 22223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-iota 5993  df-fun 6032  df-fn 6033  df-fv 6038  df-ust 22224
This theorem is referenced by:  ustbas  22251  utopval  22256  tusval  22290  ucnval  22301  iscfilu  22312
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