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Theorem ustuqtoplem 21953
Description: Lemma for ustuqtop 21960. (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 473 . . . . . . . . . . 11 ((𝑝 = 𝑞𝑣𝑈) → 𝑝 = 𝑞)
32sneqd 4160 . . . . . . . . . 10 ((𝑝 = 𝑞𝑣𝑈) → {𝑝} = {𝑞})
43imaeq2d 5425 . . . . . . . . 9 ((𝑝 = 𝑞𝑣𝑈) → (𝑣 “ {𝑝}) = (𝑣 “ {𝑞}))
54mpteq2dva 4704 . . . . . . . 8 (𝑝 = 𝑞 → (𝑣𝑈 ↦ (𝑣 “ {𝑝})) = (𝑣𝑈 ↦ (𝑣 “ {𝑞})))
65rneqd 5313 . . . . . . 7 (𝑝 = 𝑞 → ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))
76cbvmptv 4710 . . . . . 6 (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝}))) = (𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))
81, 7eqtri 2643 . . . . 5 𝑁 = (𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})))
98a1i 11 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → 𝑁 = (𝑞𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑞}))))
10 simpr2 1066 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑃𝑋𝑞 = 𝑃𝑣𝑈)) → 𝑞 = 𝑃)
1110sneqd 4160 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑃𝑋𝑞 = 𝑃𝑣𝑈)) → {𝑞} = {𝑃})
1211imaeq2d 5425 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑃𝑋𝑞 = 𝑃𝑣𝑈)) → (𝑣 “ {𝑞}) = (𝑣 “ {𝑃}))
13123anassrs 1287 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) ∧ 𝑞 = 𝑃) ∧ 𝑣𝑈) → (𝑣 “ {𝑞}) = (𝑣 “ {𝑃}))
1413mpteq2dva 4704 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) ∧ 𝑞 = 𝑃) → (𝑣𝑈 ↦ (𝑣 “ {𝑞})) = (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
1514rneqd 5313 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) ∧ 𝑞 = 𝑃) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑞})) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
16 simpr 477 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → 𝑃𝑋)
17 mptexg 6438 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
18 rnexg 7045 . . . . . 6 ((𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V → ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
1917, 18syl 17 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
2019adantr 481 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
219, 15, 16, 20fvmptd 6245 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → (𝑁𝑃) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
2221eleq2d 2684 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → (𝐴 ∈ (𝑁𝑃) ↔ 𝐴 ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃}))))
23 imaeq1 5420 . . . 4 (𝑣 = 𝑤 → (𝑣 “ {𝑃}) = (𝑤 “ {𝑃}))
2423cbvmptv 4710 . . 3 (𝑣𝑈 ↦ (𝑣 “ {𝑃})) = (𝑤𝑈 ↦ (𝑤 “ {𝑃}))
2524elrnmpt 5332 . 2 (𝐴𝑉 → (𝐴 ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ↔ ∃𝑤𝑈 𝐴 = (𝑤 “ {𝑃})))
2622, 25sylan9bb 735 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 384  w3a 1036   = wceq 1480  wcel 1987  wrex 2908  Vcvv 3186  {csn 4148  cmpt 4673  ran crn 5075  cima 5077  cfv 5847  UnifOncust 21913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855
This theorem is referenced by:  ustuqtop1  21955  ustuqtop2  21956  ustuqtop3  21957  ustuqtop4  21958  ustuqtop5  21959  utopsnneiplem  21961
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