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Theorem ustbas 24083
Description: Recover the base of an uniform structure π‘ˆ. βˆͺ ran UnifOn is to UnifOn what Top is to TopOn. (Contributed by Thierry Arnoux, 16-Nov-2017.)
Hypothesis
Ref Expression
ustbas.1 𝑋 = dom βˆͺ π‘ˆ
Assertion
Ref Expression
ustbas (π‘ˆ ∈ βˆͺ ran UnifOn ↔ π‘ˆ ∈ (UnifOnβ€˜π‘‹))

Proof of Theorem ustbas
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 ustfn 24057 . . . 4 UnifOn Fn V
2 fnfun 6642 . . . 4 (UnifOn Fn V β†’ Fun UnifOn)
3 elunirn 7245 . . . 4 (Fun UnifOn β†’ (π‘ˆ ∈ βˆͺ ran UnifOn ↔ βˆƒπ‘₯ ∈ dom UnifOnπ‘ˆ ∈ (UnifOnβ€˜π‘₯)))
41, 2, 3mp2b 10 . . 3 (π‘ˆ ∈ βˆͺ ran UnifOn ↔ βˆƒπ‘₯ ∈ dom UnifOnπ‘ˆ ∈ (UnifOnβ€˜π‘₯))
5 ustbas2 24081 . . . . . . . 8 (π‘ˆ ∈ (UnifOnβ€˜π‘₯) β†’ π‘₯ = dom βˆͺ π‘ˆ)
6 ustbas.1 . . . . . . . 8 𝑋 = dom βˆͺ π‘ˆ
75, 6eqtr4di 2784 . . . . . . 7 (π‘ˆ ∈ (UnifOnβ€˜π‘₯) β†’ π‘₯ = 𝑋)
87fveq2d 6888 . . . . . 6 (π‘ˆ ∈ (UnifOnβ€˜π‘₯) β†’ (UnifOnβ€˜π‘₯) = (UnifOnβ€˜π‘‹))
98eleq2d 2813 . . . . 5 (π‘ˆ ∈ (UnifOnβ€˜π‘₯) β†’ (π‘ˆ ∈ (UnifOnβ€˜π‘₯) ↔ π‘ˆ ∈ (UnifOnβ€˜π‘‹)))
109ibi 267 . . . 4 (π‘ˆ ∈ (UnifOnβ€˜π‘₯) β†’ π‘ˆ ∈ (UnifOnβ€˜π‘‹))
1110rexlimivw 3145 . . 3 (βˆƒπ‘₯ ∈ dom UnifOnπ‘ˆ ∈ (UnifOnβ€˜π‘₯) β†’ π‘ˆ ∈ (UnifOnβ€˜π‘‹))
124, 11sylbi 216 . 2 (π‘ˆ ∈ βˆͺ ran UnifOn β†’ π‘ˆ ∈ (UnifOnβ€˜π‘‹))
13 elfvunirn 6916 . 2 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ π‘ˆ ∈ βˆͺ ran UnifOn)
1412, 13impbii 208 1 (π‘ˆ ∈ βˆͺ ran UnifOn ↔ π‘ˆ ∈ (UnifOnβ€˜π‘‹))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3064  Vcvv 3468  βˆͺ cuni 4902  dom cdm 5669  ran crn 5670  Fun wfun 6530   Fn wfn 6531  β€˜cfv 6536  UnifOncust 24055
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-iota 6488  df-fun 6538  df-fn 6539  df-fv 6544  df-ust 24056
This theorem is referenced by: (None)
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