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Theorem ustbas2 24174
Description: Second direction for ustbas 24176. (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 5932 . 2 dom (𝑋 × 𝑋) = 𝑋
2 ustbasel 24155 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ∈ 𝑈)
3 elssuni 4941 . . . . 5 ((𝑋 × 𝑋) ∈ 𝑈 → (𝑋 × 𝑋) ⊆ 𝑈)
42, 3syl 17 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ⊆ 𝑈)
5 elfvex 6934 . . . . . . . . 9 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
6 isust 24152 . . . . . . . . 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 266 . . . . . . 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 1139 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ⊆ 𝒫 (𝑋 × 𝑋))
109unissd 4919 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 𝒫 (𝑋 × 𝑋))
11 unipw 5452 . . . . 5 𝒫 (𝑋 × 𝑋) = (𝑋 × 𝑋)
1210, 11sseqtrdi 4027 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ⊆ (𝑋 × 𝑋))
134, 12eqssd 3994 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) = 𝑈)
1413dmeqd 5908 . 2 (𝑈 ∈ (UnifOn‘𝑋) → dom (𝑋 × 𝑋) = dom 𝑈)
151, 14eqtr3id 2779 1 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1084   = wceq 1533  wcel 2098  wral 3050  wrex 3059  Vcvv 3461  cin 3943  wss 3944  𝒫 cpw 4604   cuni 4909   I cid 5575   × cxp 5676  ccnv 5677  dom cdm 5678  cres 5680  ccom 5682  cfv 6549  UnifOncust 24148
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 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-res 5690  df-iota 6501  df-fun 6551  df-fv 6557  df-ust 24149
This theorem is referenced by:  ustbas  24176  utopval  24181  tuslem  24215  tuslemOLD  24216  ucnval  24226  iscfilu  24237
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