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Theorem ustuqtop0 24265
Description: Lemma for ustuqtop 24271. (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 24253 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣𝑈𝑝𝑋) → (𝑣 “ {𝑝}) ⊆ 𝑋)
213expa 1117 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣𝑈) ∧ 𝑝𝑋) → (𝑣 “ {𝑝}) ⊆ 𝑋)
32an32s 652 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑣𝑈) → (𝑣 “ {𝑝}) ⊆ 𝑋)
4 vex 3482 . . . . . . . 8 𝑣 ∈ V
54imaex 7937 . . . . . . 7 (𝑣 “ {𝑝}) ∈ V
65elpw 4609 . . . . . 6 ((𝑣 “ {𝑝}) ∈ 𝒫 𝑋 ↔ (𝑣 “ {𝑝}) ⊆ 𝑋)
73, 6sylibr 234 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑣𝑈) → (𝑣 “ {𝑝}) ∈ 𝒫 𝑋)
87ralrimiva 3144 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → ∀𝑣𝑈 (𝑣 “ {𝑝}) ∈ 𝒫 𝑋)
9 eqid 2735 . . . . 5 (𝑣𝑈 ↦ (𝑣 “ {𝑝})) = (𝑣𝑈 ↦ (𝑣 “ {𝑝}))
109rnmptss 7143 . . . 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 7241 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V)
13 rnexg 7925 . . . . 5 ((𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V → ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V)
14 elpwg 4608 . . . . 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 480 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ 𝒫 𝒫 𝑋 ↔ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ⊆ 𝒫 𝑋))
1711, 16mpbird 257 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ 𝒫 𝒫 𝑋)
18 utopustuq.1 . 2 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
1917, 18fmptd 7134 1 (𝑈 ∈ (UnifOn‘𝑋) → 𝑁:𝑋⟶𝒫 𝒫 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  wss 3963  𝒫 cpw 4605  {csn 4631  cmpt 5231  ran crn 5690  cima 5692  wf 6559  cfv 6563  UnifOncust 24224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ust 24225
This theorem is referenced by:  ustuqtop  24271  utopsnneiplem  24272
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