Theorem ustbas2 24419

Description: Second direction for ustbas 24421. (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.

Step	Hyp	Ref	Expression
1		dmxpid 5925	. 2 ⊢ dom (𝑋 × 𝑋) = 𝑋
2		ustbasel 24401	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ∈ 𝑈)
3		elssuni 4909	. . . . 5 ⊢ ((𝑋 × 𝑋) ∈ 𝑈 → (𝑋 × 𝑋) ⊆ ∪ 𝑈)
4	2, 3	syl 18	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ⊆ ∪ 𝑈)
5		elfvex 6923	. . . . . . . . 9 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
6		isust 24398	. . . . . . . . 9 ⊢ (𝑋 ∈ V → (𝑈 ∈ (UnifOn‘𝑋) ↔ (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ 𝑈 ∧ ∀𝑣 ∈ 𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑈) ∧ ∀𝑤 ∈ 𝑈 (𝑣 ∩ 𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑣)))))
7	5, 6	syl 18	. . . . . . . 8 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑈 ∈ (UnifOn‘𝑋) ↔ (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ 𝑈 ∧ ∀𝑣 ∈ 𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑈) ∧ ∀𝑤 ∈ 𝑈 (𝑣 ∩ 𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑣)))))
8	7	ibi 270	. . . . . . 7 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ 𝑈 ∧ ∀𝑣 ∈ 𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑈) ∧ ∀𝑤 ∈ 𝑈 (𝑣 ∩ 𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑣))))
9	8	simp1d 1160	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ⊆ 𝒫 (𝑋 × 𝑋))
10	9	unissd 4887	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∪ 𝑈 ⊆ ∪ 𝒫 (𝑋 × 𝑋))
11		unipw 5436	. . . . 5 ⊢ ∪ 𝒫 (𝑋 × 𝑋) = (𝑋 × 𝑋)
12	10, 11	sseqtrdi 3980	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∪ 𝑈 ⊆ (𝑋 × 𝑋))
13	4, 12	eqssd 3957	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) = ∪ 𝑈)
14	13	dmeqd 5900	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → dom (𝑋 × 𝑋) = dom ∪ 𝑈)
15	1, 14	eqtr3id 2815	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom ∪ 𝑈)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3082  ∃wrex 3092  Vcvv 3458   ∩ cin 3907   ⊆ wss 3908  𝒫 cpw 4567  ∪ cuni 4877   I cid 5560   × cxp 5664  ^◡ccnv 5665  dom cdm 5666   ↾ cres 5668   ∘ ccom 5670  ‘cfv 6543  UnifOncust 24394

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-res 5678  df-iota 6499  df-fun 6545  df-fv 6551  df-ust 24395

This theorem is used by:  ustbas  24421  utopval  24426  tuslem  24460  ucnval  24470  iscfilu  24481