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Theorem ustuqtop0 22371
Description: Lemma for ustuqtop 22377. (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 22359 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣𝑈𝑝𝑋) → (𝑣 “ {𝑝}) ⊆ 𝑋)
213expa 1148 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣𝑈) ∧ 𝑝𝑋) → (𝑣 “ {𝑝}) ⊆ 𝑋)
32an32s 643 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑣𝑈) → (𝑣 “ {𝑝}) ⊆ 𝑋)
4 vex 3389 . . . . . . . 8 𝑣 ∈ V
54imaex 7340 . . . . . . 7 (𝑣 “ {𝑝}) ∈ V
65elpw 4356 . . . . . 6 ((𝑣 “ {𝑝}) ∈ 𝒫 𝑋 ↔ (𝑣 “ {𝑝}) ⊆ 𝑋)
73, 6sylibr 226 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑣𝑈) → (𝑣 “ {𝑝}) ∈ 𝒫 𝑋)
87ralrimiva 3148 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → ∀𝑣𝑈 (𝑣 “ {𝑝}) ∈ 𝒫 𝑋)
9 eqid 2800 . . . . 5 (𝑣𝑈 ↦ (𝑣 “ {𝑝})) = (𝑣𝑈 ↦ (𝑣 “ {𝑝}))
109rnmptss 6619 . . . 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 6714 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V)
13 rnexg 7333 . . . . 5 ((𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V → ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V)
14 elpwg 4358 . . . . 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 473 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ 𝒫 𝒫 𝑋 ↔ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ⊆ 𝒫 𝑋))
1711, 16mpbird 249 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ 𝒫 𝒫 𝑋)
18 utopustuq.1 . 2 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
1917, 18fmptd 6611 1 (𝑈 ∈ (UnifOn‘𝑋) → 𝑁:𝑋⟶𝒫 𝒫 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wral 3090  Vcvv 3386  wss 3770  𝒫 cpw 4350  {csn 4369  cmpt 4923  ran crn 5314  cima 5316  wf 6098  cfv 6102  UnifOncust 22330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-rep 4965  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-iun 4713  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-f1 6107  df-fo 6108  df-f1o 6109  df-fv 6110  df-ust 22331
This theorem is referenced by:  ustuqtop  22377  utopsnneiplem  22378
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