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Theorem ustuqtop0 22216
Description: Lemma for ustuqtop 22222. (Contributed by Thierry Arnoux, 11-Jan-2018.)
Hypothesis
Ref Expression
utopustuq.1 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
Assertion
Ref Expression
ustuqtop0 (𝑈 ∈ (UnifOn‘𝑋) → 𝑁:𝑋⟶𝒫 𝒫 𝑋)
Distinct variable groups:   𝑣,𝑝,𝑈   𝑋,𝑝,𝑣   𝑁,𝑝
Allowed substitution hint:   𝑁(𝑣)

Proof of Theorem ustuqtop0
StepHypRef Expression
1 ustimasn 22204 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣𝑈𝑝𝑋) → (𝑣 “ {𝑝}) ⊆ 𝑋)
213expa 1111 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣𝑈) ∧ 𝑝𝑋) → (𝑣 “ {𝑝}) ⊆ 𝑋)
32an32s 881 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑣𝑈) → (𝑣 “ {𝑝}) ⊆ 𝑋)
4 vex 3331 . . . . . . . 8 𝑣 ∈ V
54imaex 7257 . . . . . . 7 (𝑣 “ {𝑝}) ∈ V
65elpw 4296 . . . . . 6 ((𝑣 “ {𝑝}) ∈ 𝒫 𝑋 ↔ (𝑣 “ {𝑝}) ⊆ 𝑋)
73, 6sylibr 224 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑣𝑈) → (𝑣 “ {𝑝}) ∈ 𝒫 𝑋)
87ralrimiva 3092 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → ∀𝑣𝑈 (𝑣 “ {𝑝}) ∈ 𝒫 𝑋)
9 eqid 2748 . . . . 5 (𝑣𝑈 ↦ (𝑣 “ {𝑝})) = (𝑣𝑈 ↦ (𝑣 “ {𝑝}))
109rnmptss 6543 . . . 4 (∀𝑣𝑈 (𝑣 “ {𝑝}) ∈ 𝒫 𝑋 → ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ⊆ 𝒫 𝑋)
118, 10syl 17 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ⊆ 𝒫 𝑋)
12 mptexg 6636 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V)
13 rnexg 7251 . . . . 5 ((𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V → ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V)
14 elpwg 4298 . . . . 5 (ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V → (ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ 𝒫 𝒫 𝑋 ↔ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ⊆ 𝒫 𝑋))
1512, 13, 143syl 18 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ 𝒫 𝒫 𝑋 ↔ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ⊆ 𝒫 𝑋))
1615adantr 472 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ 𝒫 𝒫 𝑋 ↔ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ⊆ 𝒫 𝑋))
1711, 16mpbird 247 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ 𝒫 𝒫 𝑋)
18 utopustuq.1 . 2 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
1917, 18fmptd 6536 1 (𝑈 ∈ (UnifOn‘𝑋) → 𝑁:𝑋⟶𝒫 𝒫 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1620  wcel 2127  wral 3038  Vcvv 3328  wss 3703  𝒫 cpw 4290  {csn 4309  cmpt 4869  ran crn 5255  cima 5257  wf 6033  cfv 6037  UnifOncust 22175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-ust 22176
This theorem is referenced by:  ustuqtop  22222  utopsnneiplem  22223
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