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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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