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Theorem ustbas2 22820
Description: Second direction for ustbas 22822. (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 5781 . 2 dom (𝑋 × 𝑋) = 𝑋
2 ustbasel 22801 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ∈ 𝑈)
3 elssuni 4849 . . . . 5 ((𝑋 × 𝑋) ∈ 𝑈 → (𝑋 × 𝑋) ⊆ 𝑈)
42, 3syl 17 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ⊆ 𝑈)
5 elfvex 6684 . . . . . . . . 9 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
6 isust 22798 . . . . . . . . 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 270 . . . . . . 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 4829 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 𝒫 (𝑋 × 𝑋))
11 unipw 5324 . . . . 5 𝒫 (𝑋 × 𝑋) = (𝑋 × 𝑋)
1210, 11sseqtrdi 4001 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ⊆ (𝑋 × 𝑋))
134, 12eqssd 3968 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) = 𝑈)
1413dmeqd 5755 . 2 (𝑈 ∈ (UnifOn‘𝑋) → dom (𝑋 × 𝑋) = dom 𝑈)
151, 14syl5eqr 2873 1 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  w3a 1084   = wceq 1538  wcel 2115  wral 3132  wrex 3133  Vcvv 3479  cin 3917  wss 3918  𝒫 cpw 4520   cuni 4819   I cid 5440   × cxp 5534  ccnv 5535  dom cdm 5536  cres 5538  ccom 5540  cfv 6336  UnifOncust 22794
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-res 5548  df-iota 6295  df-fun 6338  df-fv 6344  df-ust 22795
This theorem is referenced by:  ustbas  22822  utopval  22827  tuslem  22862  ucnval  22872  iscfilu  22883
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