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Theorem ustbas2 24235
Description: Second direction for ustbas 24237. (Contributed by Thierry Arnoux, 16-Nov-2017.)
Assertion
Ref Expression
ustbas2 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom 𝑈)

Proof of Theorem ustbas2
Dummy variables 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmxpid 5940 . 2 dom (𝑋 × 𝑋) = 𝑋
2 ustbasel 24216 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ∈ 𝑈)
3 elssuni 4936 . . . . 5 ((𝑋 × 𝑋) ∈ 𝑈 → (𝑋 × 𝑋) ⊆ 𝑈)
42, 3syl 17 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ⊆ 𝑈)
5 elfvex 6943 . . . . . . . . 9 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
6 isust 24213 . . . . . . . . 9 (𝑋 ∈ V → (𝑈 ∈ (UnifOn‘𝑋) ↔ (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ 𝑈 ∧ ∀𝑣𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈) ∧ ∀𝑤𝑈 (𝑣𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣𝑣𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣)))))
75, 6syl 17 . . . . . . . 8 (𝑈 ∈ (UnifOn‘𝑋) → (𝑈 ∈ (UnifOn‘𝑋) ↔ (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ 𝑈 ∧ ∀𝑣𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈) ∧ ∀𝑤𝑈 (𝑣𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣𝑣𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣)))))
87ibi 267 . . . . . . 7 (𝑈 ∈ (UnifOn‘𝑋) → (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ 𝑈 ∧ ∀𝑣𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈) ∧ ∀𝑤𝑈 (𝑣𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣𝑣𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣))))
98simp1d 1142 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ⊆ 𝒫 (𝑋 × 𝑋))
109unissd 4916 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 𝒫 (𝑋 × 𝑋))
11 unipw 5454 . . . . 5 𝒫 (𝑋 × 𝑋) = (𝑋 × 𝑋)
1210, 11sseqtrdi 4023 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ⊆ (𝑋 × 𝑋))
134, 12eqssd 4000 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) = 𝑈)
1413dmeqd 5915 . 2 (𝑈 ∈ (UnifOn‘𝑋) → dom (𝑋 × 𝑋) = dom 𝑈)
151, 14eqtr3id 2790 1 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1086   = wceq 1539  wcel 2107  wral 3060  wrex 3069  Vcvv 3479  cin 3949  wss 3950  𝒫 cpw 4599   cuni 4906   I cid 5576   × cxp 5682  ccnv 5683  dom cdm 5684  cres 5686  ccom 5688  cfv 6560  UnifOncust 24209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-res 5696  df-iota 6513  df-fun 6562  df-fv 6568  df-ust 24210
This theorem is referenced by:  ustbas  24237  utopval  24242  tuslem  24276  tuslemOLD  24277  ucnval  24287  iscfilu  24298
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