Theorem ustbas2 24200

Description: Second direction for ustbas 24202. (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 5879	. 2 ⊢ dom (𝑋 × 𝑋) = 𝑋
2		ustbasel 24182	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ∈ 𝑈)
3		elssuni 4882	. . . . 5 ⊢ ((𝑋 × 𝑋) ∈ 𝑈 → (𝑋 × 𝑋) ⊆ ∪ 𝑈)
4	2, 3	syl 17	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ⊆ ∪ 𝑈)
5		elfvex 6869	. . . . . . . . 9 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
6		isust 24179	. . . . . . . . 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 17	. . . . . . . 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 267	. . . . . . 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 1143	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ⊆ 𝒫 (𝑋 × 𝑋))
10	9	unissd 4861	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∪ 𝑈 ⊆ ∪ 𝒫 (𝑋 × 𝑋))
11		unipw 5397	. . . . 5 ⊢ ∪ 𝒫 (𝑋 × 𝑋) = (𝑋 × 𝑋)
12	10, 11	sseqtrdi 3963	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∪ 𝑈 ⊆ (𝑋 × 𝑋))
13	4, 12	eqssd 3940	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) = ∪ 𝑈)
14	13	dmeqd 5854	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → dom (𝑋 × 𝑋) = dom ∪ 𝑈)
15	1, 14	eqtr3id 2786	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom ∪ 𝑈)

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  Vcvv 3430   ∩ cin 3889   ⊆ wss 3890  𝒫 cpw 4542  ∪ cuni 4851   I cid 5518   × cxp 5622  ^◡ccnv 5623  dom cdm 5624   ↾ cres 5626   ∘ ccom 5628  ‘cfv 6492  UnifOncust 24175

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-res 5636  df-iota 6448  df-fun 6494  df-fv 6500  df-ust 24176

This theorem is referenced by:  ustbas  24202  utopval  24207  tuslem  24241  ucnval  24251  iscfilu  24262