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Theorem ustbas 24252
Description: Recover the base of an uniform structure 𝑈. ran UnifOn is to UnifOn what Top is to TopOn. (Contributed by Thierry Arnoux, 16-Nov-2017.)
Hypothesis
Ref Expression
ustbas.1 𝑋 = dom 𝑈
Assertion
Ref Expression
ustbas (𝑈 ran UnifOn ↔ 𝑈 ∈ (UnifOn‘𝑋))

Proof of Theorem ustbas
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ustfn 24226 . . . 4 UnifOn Fn V
2 fnfun 6669 . . . 4 (UnifOn Fn V → Fun UnifOn)
3 elunirn 7271 . . . 4 (Fun UnifOn → (𝑈 ran UnifOn ↔ ∃𝑥 ∈ dom UnifOn𝑈 ∈ (UnifOn‘𝑥)))
41, 2, 3mp2b 10 . . 3 (𝑈 ran UnifOn ↔ ∃𝑥 ∈ dom UnifOn𝑈 ∈ (UnifOn‘𝑥))
5 ustbas2 24250 . . . . . . . 8 (𝑈 ∈ (UnifOn‘𝑥) → 𝑥 = dom 𝑈)
6 ustbas.1 . . . . . . . 8 𝑋 = dom 𝑈
75, 6eqtr4di 2793 . . . . . . 7 (𝑈 ∈ (UnifOn‘𝑥) → 𝑥 = 𝑋)
87fveq2d 6911 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑥) → (UnifOn‘𝑥) = (UnifOn‘𝑋))
98eleq2d 2825 . . . . 5 (𝑈 ∈ (UnifOn‘𝑥) → (𝑈 ∈ (UnifOn‘𝑥) ↔ 𝑈 ∈ (UnifOn‘𝑋)))
109ibi 267 . . . 4 (𝑈 ∈ (UnifOn‘𝑥) → 𝑈 ∈ (UnifOn‘𝑋))
1110rexlimivw 3149 . . 3 (∃𝑥 ∈ dom UnifOn𝑈 ∈ (UnifOn‘𝑥) → 𝑈 ∈ (UnifOn‘𝑋))
124, 11sylbi 217 . 2 (𝑈 ran UnifOn → 𝑈 ∈ (UnifOn‘𝑋))
13 elfvunirn 6939 . 2 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ran UnifOn)
1412, 13impbii 209 1 (𝑈 ran UnifOn ↔ 𝑈 ∈ (UnifOn‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1537  wcel 2106  wrex 3068  Vcvv 3478   cuni 4912  dom cdm 5689  ran crn 5690  Fun wfun 6557   Fn wfn 6558  cfv 6563  UnifOncust 24224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-ust 24225
This theorem is referenced by: (None)
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