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Theorem ustuqtoplem 22240
Description: Lemma for ustuqtop 22247. (Contributed by Thierry Arnoux, 11-Jan-2018.)
Hypothesis
Ref Expression
utopustuq.1 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
Assertion
Ref Expression
ustuqtoplem (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) ∧ 𝐴𝑉) → (𝐴 ∈ (𝑁𝑃) ↔ ∃𝑤𝑈 𝐴 = (𝑤 “ {𝑃})))
Distinct variable groups:   𝑤,𝐴   𝑤,𝑣,𝑃   𝑣,𝑝,𝑤,𝑈   𝑋,𝑝,𝑣
Allowed substitution hints:   𝐴(𝑣,𝑝)   𝑃(𝑝)   𝑁(𝑤,𝑣,𝑝)   𝑉(𝑤,𝑣,𝑝)   𝑋(𝑤)

Proof of Theorem ustuqtoplem
Dummy variable 𝑞 is distinct from all other variables.
StepHypRef Expression
1 utopustuq.1 . . . . . 6 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
2 simpl 474 . . . . . . . . . . 11 ((𝑝 = 𝑞𝑣𝑈) → 𝑝 = 𝑞)
32sneqd 4329 . . . . . . . . . 10 ((𝑝 = 𝑞𝑣𝑈) → {𝑝} = {𝑞})
43imaeq2d 5620 . . . . . . . . 9 ((𝑝 = 𝑞𝑣𝑈) → (𝑣 “ {𝑝}) = (𝑣 “ {𝑞}))
54mpteq2dva 4892 . . . . . . . 8 (𝑝 = 𝑞 → (𝑣𝑈 ↦ (𝑣 “ {𝑝})) = (𝑣𝑈 ↦ (𝑣 “ {𝑞})))
65rneqd 5504 . . . . . . 7 (𝑝 = 𝑞 → ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))
76cbvmptv 4898 . . . . . 6 (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝}))) = (𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))
81, 7eqtri 2778 . . . . 5 𝑁 = (𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))
98a1i 11 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → 𝑁 = (𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞}))))
10 simpr2 1236 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑃𝑋𝑞 = 𝑃𝑣𝑈)) → 𝑞 = 𝑃)
1110sneqd 4329 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑃𝑋𝑞 = 𝑃𝑣𝑈)) → {𝑞} = {𝑃})
1211imaeq2d 5620 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑃𝑋𝑞 = 𝑃𝑣𝑈)) → (𝑣 “ {𝑞}) = (𝑣 “ {𝑃}))
13123anassrs 1454 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) ∧ 𝑞 = 𝑃) ∧ 𝑣𝑈) → (𝑣 “ {𝑞}) = (𝑣 “ {𝑃}))
1413mpteq2dva 4892 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) ∧ 𝑞 = 𝑃) → (𝑣𝑈 ↦ (𝑣 “ {𝑞})) = (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
1514rneqd 5504 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) ∧ 𝑞 = 𝑃) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
16 simpr 479 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → 𝑃𝑋)
17 mptexg 6644 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
18 rnexg 7259 . . . . . 6 ((𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V → ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
1917, 18syl 17 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
2019adantr 472 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
219, 15, 16, 20fvmptd 6446 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → (𝑁𝑃) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
2221eleq2d 2821 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → (𝐴 ∈ (𝑁𝑃) ↔ 𝐴 ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃}))))
23 imaeq1 5615 . . . 4 (𝑣 = 𝑤 → (𝑣 “ {𝑃}) = (𝑤 “ {𝑃}))
2423cbvmptv 4898 . . 3 (𝑣𝑈 ↦ (𝑣 “ {𝑃})) = (𝑤𝑈 ↦ (𝑤 “ {𝑃}))
2524elrnmpt 5523 . 2 (𝐴𝑉 → (𝐴 ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ↔ ∃𝑤𝑈 𝐴 = (𝑤 “ {𝑃})))
2622, 25sylan9bb 738 1 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) ∧ 𝐴𝑉) → (𝐴 ∈ (𝑁𝑃) ↔ ∃𝑤𝑈 𝐴 = (𝑤 “ {𝑃})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1628  wcel 2135  wrex 3047  Vcvv 3336  {csn 4317  cmpt 4877  ran crn 5263  cima 5265  cfv 6045  UnifOncust 22200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-rep 4919  ax-sep 4929  ax-nul 4937  ax-pr 5051  ax-un 7110
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-reu 3053  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4585  df-iun 4670  df-br 4801  df-opab 4861  df-mpt 4878  df-id 5170  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-f1 6050  df-fo 6051  df-f1o 6052  df-fv 6053
This theorem is referenced by:  ustuqtop1  22242  ustuqtop2  22243  ustuqtop3  22244  ustuqtop4  22245  ustuqtop5  22246  utopsnneiplem  22248
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