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Theorem isust 22200
 Description: The predicate "𝑈 is a uniform structure with base 𝑋." (Contributed by Thierry Arnoux, 15-Nov-2017.) (Revised by AV, 17-Sep-2021.)
Assertion
Ref Expression
isust (𝑋𝑉 → (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)))))
Distinct variable groups:   𝑤,𝑣,𝑈   𝑣,𝑋,𝑤
Allowed substitution hints:   𝑉(𝑤,𝑣)

Proof of Theorem isust
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 ustval 22199 . . 3 (𝑋𝑉 → (UnifOn‘𝑋) = {𝑢 ∣ (𝑢 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ 𝑢 ∧ ∀𝑣𝑢 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑢) ∧ ∀𝑤𝑢 (𝑣𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑋) ⊆ 𝑣𝑣𝑢 ∧ ∃𝑤𝑢 (𝑤𝑤) ⊆ 𝑣)))})
21eleq2d 2817 . 2 (𝑋𝑉 → (𝑈 ∈ (UnifOn‘𝑋) ↔ 𝑈 ∈ {𝑢 ∣ (𝑢 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ 𝑢 ∧ ∀𝑣𝑢 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑢) ∧ ∀𝑤𝑢 (𝑣𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑋) ⊆ 𝑣𝑣𝑢 ∧ ∃𝑤𝑢 (𝑤𝑤) ⊆ 𝑣)))}))
3 simp1 1130 . . . 4 ((𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ 𝑈 ∧ ∀𝑣𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈) ∧ ∀𝑤𝑈 (𝑣𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣𝑣𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣))) → 𝑈 ⊆ 𝒫 (𝑋 × 𝑋))
4 sqxpexg 7120 . . . . . . . 8 (𝑋𝑉 → (𝑋 × 𝑋) ∈ V)
5 pwexg 4991 . . . . . . . 8 ((𝑋 × 𝑋) ∈ V → 𝒫 (𝑋 × 𝑋) ∈ V)
64, 5syl 17 . . . . . . 7 (𝑋𝑉 → 𝒫 (𝑋 × 𝑋) ∈ V)
76adantr 472 . . . . . 6 ((𝑋𝑉𝑈 ⊆ 𝒫 (𝑋 × 𝑋)) → 𝒫 (𝑋 × 𝑋) ∈ V)
8 simpr 479 . . . . . 6 ((𝑋𝑉𝑈 ⊆ 𝒫 (𝑋 × 𝑋)) → 𝑈 ⊆ 𝒫 (𝑋 × 𝑋))
97, 8ssexd 4949 . . . . 5 ((𝑋𝑉𝑈 ⊆ 𝒫 (𝑋 × 𝑋)) → 𝑈 ∈ V)
109ex 449 . . . 4 (𝑋𝑉 → (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) → 𝑈 ∈ V))
113, 10syl5 34 . . 3 (𝑋𝑉 → ((𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ 𝑈 ∧ ∀𝑣𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈) ∧ ∀𝑤𝑈 (𝑣𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣𝑣𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣))) → 𝑈 ∈ V))
12 sseq1 3759 . . . . 5 (𝑢 = 𝑈 → (𝑢 ⊆ 𝒫 (𝑋 × 𝑋) ↔ 𝑈 ⊆ 𝒫 (𝑋 × 𝑋)))
13 eleq2 2820 . . . . 5 (𝑢 = 𝑈 → ((𝑋 × 𝑋) ∈ 𝑢 ↔ (𝑋 × 𝑋) ∈ 𝑈))
14 eleq2 2820 . . . . . . . . 9 (𝑢 = 𝑈 → (𝑤𝑢𝑤𝑈))
1514imbi2d 329 . . . . . . . 8 (𝑢 = 𝑈 → ((𝑣𝑤𝑤𝑢) ↔ (𝑣𝑤𝑤𝑈)))
1615ralbidv 3116 . . . . . . 7 (𝑢 = 𝑈 → (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑢) ↔ ∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈)))
17 eleq2 2820 . . . . . . . 8 (𝑢 = 𝑈 → ((𝑣𝑤) ∈ 𝑢 ↔ (𝑣𝑤) ∈ 𝑈))
1817raleqbi1dv 3277 . . . . . . 7 (𝑢 = 𝑈 → (∀𝑤𝑢 (𝑣𝑤) ∈ 𝑢 ↔ ∀𝑤𝑈 (𝑣𝑤) ∈ 𝑈))
19 eleq2 2820 . . . . . . . 8 (𝑢 = 𝑈 → (𝑣𝑢𝑣𝑈))
20 rexeq 3270 . . . . . . . 8 (𝑢 = 𝑈 → (∃𝑤𝑢 (𝑤𝑤) ⊆ 𝑣 ↔ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣))
2119, 203anbi23d 1543 . . . . . . 7 (𝑢 = 𝑈 → ((( I ↾ 𝑋) ⊆ 𝑣𝑣𝑢 ∧ ∃𝑤𝑢 (𝑤𝑤) ⊆ 𝑣) ↔ (( I ↾ 𝑋) ⊆ 𝑣𝑣𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣)))
2216, 18, 213anbi123d 1540 . . . . . 6 (𝑢 = 𝑈 → ((∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑢) ∧ ∀𝑤𝑢 (𝑣𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑋) ⊆ 𝑣𝑣𝑢 ∧ ∃𝑤𝑢 (𝑤𝑤) ⊆ 𝑣)) ↔ (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈) ∧ ∀𝑤𝑈 (𝑣𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣𝑣𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣))))
2322raleqbi1dv 3277 . . . . 5 (𝑢 = 𝑈 → (∀𝑣𝑢 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑢) ∧ ∀𝑤𝑢 (𝑣𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑋) ⊆ 𝑣𝑣𝑢 ∧ ∃𝑤𝑢 (𝑤𝑤) ⊆ 𝑣)) ↔ ∀𝑣𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈) ∧ ∀𝑤𝑈 (𝑣𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣𝑣𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣))))
2412, 13, 233anbi123d 1540 . . . 4 (𝑢 = 𝑈 → ((𝑢 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ 𝑢 ∧ ∀𝑣𝑢 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑢) ∧ ∀𝑤𝑢 (𝑣𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑋) ⊆ 𝑣𝑣𝑢 ∧ ∃𝑤𝑢 (𝑤𝑤) ⊆ 𝑣))) ↔ (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ 𝑈 ∧ ∀𝑣𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈) ∧ ∀𝑤𝑈 (𝑣𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣𝑣𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣)))))
2524elab3g 3489 . . 3 (((𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ 𝑈 ∧ ∀𝑣𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈) ∧ ∀𝑤𝑈 (𝑣𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣𝑣𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣))) → 𝑈 ∈ V) → (𝑈 ∈ {𝑢 ∣ (𝑢 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ 𝑢 ∧ ∀𝑣𝑢 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑢) ∧ ∀𝑤𝑢 (𝑣𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑋) ⊆ 𝑣𝑣𝑢 ∧ ∃𝑤𝑢 (𝑤𝑤) ⊆ 𝑣)))} ↔ (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ 𝑈 ∧ ∀𝑣𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈) ∧ ∀𝑤𝑈 (𝑣𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣𝑣𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣)))))
2611, 25syl 17 . 2 (𝑋𝑉 → (𝑈 ∈ {𝑢 ∣ (𝑢 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ 𝑢 ∧ ∀𝑣𝑢 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑢) ∧ ∀𝑤𝑢 (𝑣𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑋) ⊆ 𝑣𝑣𝑢 ∧ ∃𝑤𝑢 (𝑤𝑤) ⊆ 𝑣)))} ↔ (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ 𝑈 ∧ ∀𝑣𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈) ∧ ∀𝑤𝑈 (𝑣𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣𝑣𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣)))))
272, 26bitrd 268 1 (𝑋𝑉 → (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)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1072   = wceq 1624   ∈ wcel 2131  {cab 2738  ∀wral 3042  ∃wrex 3043  Vcvv 3332   ∩ cin 3706   ⊆ wss 3707  𝒫 cpw 4294   I cid 5165   × cxp 5256  ◡ccnv 5257   ↾ cres 5260   ∘ ccom 5262  ‘cfv 6041  UnifOncust 22196 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-res 5270  df-iota 6004  df-fun 6043  df-fv 6049  df-ust 22197 This theorem is referenced by:  ustssxp  22201  ustssel  22202  ustbasel  22203  ustincl  22204  ustdiag  22205  ustinvel  22206  ustexhalf  22207  ustfilxp  22209  ust0  22216  ustbas2  22222  trust  22226  metust  22556
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