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Theorem ustbas2 21839
Description: Second direction for ustbas 21841. (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 5266 . 2 dom (𝑋 × 𝑋) = 𝑋
2 ustbasel 21820 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ∈ 𝑈)
3 elssuni 4403 . . . . 5 ((𝑋 × 𝑋) ∈ 𝑈 → (𝑋 × 𝑋) ⊆ 𝑈)
42, 3syl 17 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ⊆ 𝑈)
5 elfvex 6131 . . . . . . . . 9 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
6 isust 21817 . . . . . . . . 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 255 . . . . . . 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 1066 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ⊆ 𝒫 (𝑋 × 𝑋))
109unissd 4398 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 𝒫 (𝑋 × 𝑋))
11 unipw 4845 . . . . 5 𝒫 (𝑋 × 𝑋) = (𝑋 × 𝑋)
1210, 11syl6sseq 3614 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ⊆ (𝑋 × 𝑋))
134, 12eqssd 3585 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) = 𝑈)
1413dmeqd 5248 . 2 (𝑈 ∈ (UnifOn‘𝑋) → dom (𝑋 × 𝑋) = dom 𝑈)
151, 14syl5eqr 2658 1 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  w3a 1031   = wceq 1475  wcel 1977  wral 2896  wrex 2897  Vcvv 3173  cin 3539  wss 3540  𝒫 cpw 4108   cuni 4372   I cid 4948   × cxp 5036  ccnv 5037  dom cdm 5038  cres 5040  ccom 5042  cfv 5804  UnifOncust 21813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-res 5050  df-iota 5768  df-fun 5806  df-fv 5812  df-ust 21814
This theorem is referenced by:  ustbas  21841  utopval  21846  tuslem  21881  ucnval  21891  iscfilu  21902
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