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Theorem ustuqtoplem 24235
Description: Lemma for ustuqtop 24242. (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 . . . . 5 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
2 simpl 481 . . . . . . . . . 10 ((𝑝 = 𝑞𝑣𝑈) → 𝑝 = 𝑞)
32sneqd 4645 . . . . . . . . 9 ((𝑝 = 𝑞𝑣𝑈) → {𝑝} = {𝑞})
43imaeq2d 6069 . . . . . . . 8 ((𝑝 = 𝑞𝑣𝑈) → (𝑣 “ {𝑝}) = (𝑣 “ {𝑞}))
54mpteq2dva 5253 . . . . . . 7 (𝑝 = 𝑞 → (𝑣𝑈 ↦ (𝑣 “ {𝑝})) = (𝑣𝑈 ↦ (𝑣 “ {𝑞})))
65rneqd 5944 . . . . . 6 (𝑝 = 𝑞 → ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))
76cbvmptv 5266 . . . . 5 (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝}))) = (𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))
81, 7eqtri 2754 . . . 4 𝑁 = (𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))
9 simpr2 1192 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑃𝑋𝑞 = 𝑃𝑣𝑈)) → 𝑞 = 𝑃)
109sneqd 4645 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑃𝑋𝑞 = 𝑃𝑣𝑈)) → {𝑞} = {𝑃})
1110imaeq2d 6069 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑃𝑋𝑞 = 𝑃𝑣𝑈)) → (𝑣 “ {𝑞}) = (𝑣 “ {𝑃}))
12113anassrs 1357 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) ∧ 𝑞 = 𝑃) ∧ 𝑣𝑈) → (𝑣 “ {𝑞}) = (𝑣 “ {𝑃}))
1312mpteq2dva 5253 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) ∧ 𝑞 = 𝑃) → (𝑣𝑈 ↦ (𝑣 “ {𝑞})) = (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
1413rneqd 5944 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) ∧ 𝑞 = 𝑃) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
15 simpr 483 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → 𝑃𝑋)
16 mptexg 7238 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
17 rnexg 7915 . . . . . 6 ((𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V → ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
1816, 17syl 17 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
1918adantr 479 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
208, 14, 15, 19fvmptd2 7017 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → (𝑁𝑃) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
2120eleq2d 2812 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → (𝐴 ∈ (𝑁𝑃) ↔ 𝐴 ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃}))))
22 imaeq1 6064 . . . 4 (𝑣 = 𝑤 → (𝑣 “ {𝑃}) = (𝑤 “ {𝑃}))
2322cbvmptv 5266 . . 3 (𝑣𝑈 ↦ (𝑣 “ {𝑃})) = (𝑤𝑈 ↦ (𝑤 “ {𝑃}))
2423elrnmpt 5962 . 2 (𝐴𝑉 → (𝐴 ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ↔ ∃𝑤𝑈 𝐴 = (𝑤 “ {𝑃})))
2521, 24sylan9bb 508 1 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) ∧ 𝐴𝑉) → (𝐴 ∈ (𝑁𝑃) ↔ ∃𝑤𝑈 𝐴 = (𝑤 “ {𝑃})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1534  wcel 2099  wrex 3060  Vcvv 3462  {csn 4633  cmpt 5236  ran crn 5683  cima 5685  cfv 6554  UnifOncust 24195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562
This theorem is referenced by:  ustuqtop1  24237  ustuqtop2  24238  ustuqtop3  24239  ustuqtop4  24240  ustuqtop5  24241  utopsnneiplem  24243
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