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Theorem ustuqtoplem 22263
Description: Lemma for ustuqtop 22270. (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 468 . . . . . . . . . . 11 ((𝑝 = 𝑞𝑣𝑈) → 𝑝 = 𝑞)
32sneqd 4328 . . . . . . . . . 10 ((𝑝 = 𝑞𝑣𝑈) → {𝑝} = {𝑞})
43imaeq2d 5607 . . . . . . . . 9 ((𝑝 = 𝑞𝑣𝑈) → (𝑣 “ {𝑝}) = (𝑣 “ {𝑞}))
54mpteq2dva 4878 . . . . . . . 8 (𝑝 = 𝑞 → (𝑣𝑈 ↦ (𝑣 “ {𝑝})) = (𝑣𝑈 ↦ (𝑣 “ {𝑞})))
65rneqd 5491 . . . . . . 7 (𝑝 = 𝑞 → ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))
76cbvmptv 4884 . . . . . 6 (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝}))) = (𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))
81, 7eqtri 2793 . . . . 5 𝑁 = (𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))
98a1i 11 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → 𝑁 = (𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞}))))
10 simpr2 1235 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑃𝑋𝑞 = 𝑃𝑣𝑈)) → 𝑞 = 𝑃)
1110sneqd 4328 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑃𝑋𝑞 = 𝑃𝑣𝑈)) → {𝑞} = {𝑃})
1211imaeq2d 5607 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑃𝑋𝑞 = 𝑃𝑣𝑈)) → (𝑣 “ {𝑞}) = (𝑣 “ {𝑃}))
13123anassrs 1453 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) ∧ 𝑞 = 𝑃) ∧ 𝑣𝑈) → (𝑣 “ {𝑞}) = (𝑣 “ {𝑃}))
1413mpteq2dva 4878 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) ∧ 𝑞 = 𝑃) → (𝑣𝑈 ↦ (𝑣 “ {𝑞})) = (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
1514rneqd 5491 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) ∧ 𝑞 = 𝑃) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
16 simpr 471 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → 𝑃𝑋)
17 mptexg 6628 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
18 rnexg 7245 . . . . . 6 ((𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V → ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
1917, 18syl 17 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
2019adantr 466 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
219, 15, 16, 20fvmptd 6430 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → (𝑁𝑃) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
2221eleq2d 2836 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → (𝐴 ∈ (𝑁𝑃) ↔ 𝐴 ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃}))))
23 imaeq1 5602 . . . 4 (𝑣 = 𝑤 → (𝑣 “ {𝑃}) = (𝑤 “ {𝑃}))
2423cbvmptv 4884 . . 3 (𝑣𝑈 ↦ (𝑣 “ {𝑃})) = (𝑤𝑈 ↦ (𝑤 “ {𝑃}))
2524elrnmpt 5510 . 2 (𝐴𝑉 → (𝐴 ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ↔ ∃𝑤𝑈 𝐴 = (𝑤 “ {𝑃})))
2622, 25sylan9bb 499 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 382  w3a 1071   = wceq 1631  wcel 2145  wrex 3062  Vcvv 3351  {csn 4316  cmpt 4863  ran crn 5250  cima 5252  cfv 6031  UnifOncust 22223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039
This theorem is referenced by:  ustuqtop1  22265  ustuqtop2  22266  ustuqtop3  22267  ustuqtop4  22268  ustuqtop5  22269  utopsnneiplem  22271
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