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Theorem ustbas 22831
 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 22805 . . . 4 UnifOn Fn V
2 fnfun 6432 . . . 4 (UnifOn Fn V → Fun UnifOn)
3 elunirn 6993 . . . 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 22829 . . . . . . . 8 (𝑈 ∈ (UnifOn‘𝑥) → 𝑥 = dom 𝑈)
6 ustbas.1 . . . . . . . 8 𝑋 = dom 𝑈
75, 6eqtr4di 2875 . . . . . . 7 (𝑈 ∈ (UnifOn‘𝑥) → 𝑥 = 𝑋)
87fveq2d 6656 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑥) → (UnifOn‘𝑥) = (UnifOn‘𝑋))
98eleq2d 2899 . . . . 5 (𝑈 ∈ (UnifOn‘𝑥) → (𝑈 ∈ (UnifOn‘𝑥) ↔ 𝑈 ∈ (UnifOn‘𝑋)))
109ibi 270 . . . 4 (𝑈 ∈ (UnifOn‘𝑥) → 𝑈 ∈ (UnifOn‘𝑋))
1110rexlimivw 3268 . . 3 (∃𝑥 ∈ dom UnifOn𝑈 ∈ (UnifOn‘𝑥) → 𝑈 ∈ (UnifOn‘𝑋))
124, 11sylbi 220 . 2 (𝑈 ran UnifOn → 𝑈 ∈ (UnifOn‘𝑋))
13 elrnust 22828 . 2 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ran UnifOn)
1412, 13impbii 212 1 (𝑈 ran UnifOn ↔ 𝑈 ∈ (UnifOn‘𝑋))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538   ∈ wcel 2114  ∃wrex 3131  Vcvv 3469  ∪ cuni 4813  dom cdm 5532  ran crn 5533  Fun wfun 6328   Fn wfn 6329  ‘cfv 6334  UnifOncust 22803 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-iota 6293  df-fun 6336  df-fn 6337  df-fv 6342  df-ust 22804 This theorem is referenced by: (None)
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