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Theorem isust 24212
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 24211 . . 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 2827 . 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 1137 . . . 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 7775 . . . . . . . 8 (𝑋𝑉 → (𝑋 × 𝑋) ∈ V)
54pwexd 5379 . . . . . . 7 (𝑋𝑉 → 𝒫 (𝑋 × 𝑋) ∈ V)
65adantr 480 . . . . . 6 ((𝑋𝑉𝑈 ⊆ 𝒫 (𝑋 × 𝑋)) → 𝒫 (𝑋 × 𝑋) ∈ V)
7 simpr 484 . . . . . 6 ((𝑋𝑉𝑈 ⊆ 𝒫 (𝑋 × 𝑋)) → 𝑈 ⊆ 𝒫 (𝑋 × 𝑋))
86, 7ssexd 5324 . . . . 5 ((𝑋𝑉𝑈 ⊆ 𝒫 (𝑋 × 𝑋)) → 𝑈 ∈ V)
98ex 412 . . . 4 (𝑋𝑉 → (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) → 𝑈 ∈ V))
103, 9syl5 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))
11 sseq1 4009 . . . . 5 (𝑢 = 𝑈 → (𝑢 ⊆ 𝒫 (𝑋 × 𝑋) ↔ 𝑈 ⊆ 𝒫 (𝑋 × 𝑋)))
12 eleq2 2830 . . . . 5 (𝑢 = 𝑈 → ((𝑋 × 𝑋) ∈ 𝑢 ↔ (𝑋 × 𝑋) ∈ 𝑈))
13 eleq2 2830 . . . . . . . . 9 (𝑢 = 𝑈 → (𝑤𝑢𝑤𝑈))
1413imbi2d 340 . . . . . . . 8 (𝑢 = 𝑈 → ((𝑣𝑤𝑤𝑢) ↔ (𝑣𝑤𝑤𝑈)))
1514ralbidv 3178 . . . . . . 7 (𝑢 = 𝑈 → (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑢) ↔ ∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈)))
16 eleq2 2830 . . . . . . . 8 (𝑢 = 𝑈 → ((𝑣𝑤) ∈ 𝑢 ↔ (𝑣𝑤) ∈ 𝑈))
1716raleqbi1dv 3338 . . . . . . 7 (𝑢 = 𝑈 → (∀𝑤𝑢 (𝑣𝑤) ∈ 𝑢 ↔ ∀𝑤𝑈 (𝑣𝑤) ∈ 𝑈))
18 eleq2 2830 . . . . . . . 8 (𝑢 = 𝑈 → (𝑣𝑢𝑣𝑈))
19 rexeq 3322 . . . . . . . 8 (𝑢 = 𝑈 → (∃𝑤𝑢 (𝑤𝑤) ⊆ 𝑣 ↔ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣))
2018, 193anbi23d 1441 . . . . . . 7 (𝑢 = 𝑈 → ((( I ↾ 𝑋) ⊆ 𝑣𝑣𝑢 ∧ ∃𝑤𝑢 (𝑤𝑤) ⊆ 𝑣) ↔ (( I ↾ 𝑋) ⊆ 𝑣𝑣𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣)))
2115, 17, 203anbi123d 1438 . . . . . 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))))
2221raleqbi1dv 3338 . . . . 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))))
2311, 12, 223anbi123d 1438 . . . 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)))))
2423elab3g 3685 . . 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)))))
2510, 24syl 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)))))
262, 25bitrd 279 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 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  {cab 2714  wral 3061  wrex 3070  Vcvv 3480  cin 3950  wss 3951  𝒫 cpw 4600   I cid 5577   × cxp 5683  ccnv 5684  cres 5687  ccom 5689  cfv 6561  UnifOncust 24208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-iota 6514  df-fun 6563  df-fv 6569  df-ust 24209
This theorem is referenced by:  ustssxp  24213  ustssel  24214  ustbasel  24215  ustincl  24216  ustdiag  24217  ustinvel  24218  ustexhalf  24219  ustfilxp  24221  ust0  24228  ustbas2  24234  trust  24238  metust  24571
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