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Theorem ustbas2 21939
Description: Second direction for ustbas 21941. (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 5305 . 2 dom (𝑋 × 𝑋) = 𝑋
2 ustbasel 21920 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ∈ 𝑈)
3 elssuni 4433 . . . . 5 ((𝑋 × 𝑋) ∈ 𝑈 → (𝑋 × 𝑋) ⊆ 𝑈)
42, 3syl 17 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ⊆ 𝑈)
5 elfvex 6178 . . . . . . . . 9 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
6 isust 21917 . . . . . . . . 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 256 . . . . . . 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 1071 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ⊆ 𝒫 (𝑋 × 𝑋))
109unissd 4428 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 𝒫 (𝑋 × 𝑋))
11 unipw 4879 . . . . 5 𝒫 (𝑋 × 𝑋) = (𝑋 × 𝑋)
1210, 11syl6sseq 3630 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ⊆ (𝑋 × 𝑋))
134, 12eqssd 3600 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) = 𝑈)
1413dmeqd 5286 . 2 (𝑈 ∈ (UnifOn‘𝑋) → dom (𝑋 × 𝑋) = dom 𝑈)
151, 14syl5eqr 2669 1 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  Vcvv 3186  cin 3554  wss 3555  𝒫 cpw 4130   cuni 4402   I cid 4984   × cxp 5072  ccnv 5073  dom cdm 5074  cres 5076  ccom 5078  cfv 5847  UnifOncust 21913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-res 5086  df-iota 5810  df-fun 5849  df-fv 5855  df-ust 21914
This theorem is referenced by:  ustbas  21941  utopval  21946  tuslem  21981  ucnval  21991  iscfilu  22002
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