Theorem ustuqtoplem 23401

Description: Lemma for ustuqtop 23408. (Contributed by Thierry Arnoux, 11-Jan-2018.)

Hypothesis

Ref	Expression
utopustuq.1	⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))

Assertion

Ref	Expression
ustuqtoplem	⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝐴 ∈ 𝑉) → (𝐴 ∈ (𝑁‘𝑃) ↔ ∃𝑤 ∈ 𝑈 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.

Step	Hyp	Ref	Expression
1		utopustuq.1	. . . . 5 ⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))
2		simpl 483	. . . . . . . . . 10 ⊢ ((𝑝 = 𝑞 ∧ 𝑣 ∈ 𝑈) → 𝑝 = 𝑞)
3	2	sneqd 4573	. . . . . . . . 9 ⊢ ((𝑝 = 𝑞 ∧ 𝑣 ∈ 𝑈) → {𝑝} = {𝑞})
4	3	imaeq2d 5962	. . . . . . . 8 ⊢ ((𝑝 = 𝑞 ∧ 𝑣 ∈ 𝑈) → (𝑣 “ {𝑝}) = (𝑣 “ {𝑞}))
5	4	mpteq2dva 5173	. . . . . . 7 ⊢ (𝑝 = 𝑞 → (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) = (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))
6	5	rneqd 5840	. . . . . 6 ⊢ (𝑝 = 𝑞 → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))
7	6	cbvmptv 5186	. . . . 5 ⊢ (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝}))) = (𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))
8	1, 7	eqtri 2766	. . . 4 ⊢ 𝑁 = (𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))
9		simpr2 1194	. . . . . . . . 9 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑃 ∈ 𝑋 ∧ 𝑞 = 𝑃 ∧ 𝑣 ∈ 𝑈)) → 𝑞 = 𝑃)
10	9	sneqd 4573	. . . . . . . 8 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑃 ∈ 𝑋 ∧ 𝑞 = 𝑃 ∧ 𝑣 ∈ 𝑈)) → {𝑞} = {𝑃})
11	10	imaeq2d 5962	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑃 ∈ 𝑋 ∧ 𝑞 = 𝑃 ∧ 𝑣 ∈ 𝑈)) → (𝑣 “ {𝑞}) = (𝑣 “ {𝑃}))
12	11	3anassrs 1359	. . . . . 6 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑞 = 𝑃) ∧ 𝑣 ∈ 𝑈) → (𝑣 “ {𝑞}) = (𝑣 “ {𝑃}))
13	12	mpteq2dva 5173	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑞 = 𝑃) → (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})) = (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))
14	13	rneqd 5840	. . . 4 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑞 = 𝑃) → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))
15		simpr 485	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → 𝑃 ∈ 𝑋)
16		mptexg 7089	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
17		rnexg 7741	. . . . . 6 ⊢ ((𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ∈ V → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
18	16, 17	syl 17	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
19	18	adantr 481	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
20	8, 14, 15, 19	fvmptd2 6875	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → (𝑁‘𝑃) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))
21	20	eleq2d 2824	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → (𝐴 ∈ (𝑁‘𝑃) ↔ 𝐴 ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃}))))
22		imaeq1 5957	. . . 4 ⊢ (𝑣 = 𝑤 → (𝑣 “ {𝑃}) = (𝑤 “ {𝑃}))
23	22	cbvmptv 5186	. . 3 ⊢ (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) = (𝑤 ∈ 𝑈 ↦ (𝑤 “ {𝑃}))
24	23	elrnmpt 5858	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ↔ ∃𝑤 ∈ 𝑈 𝐴 = (𝑤 “ {𝑃})))
25	21, 24	sylan9bb 510	1 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝐴 ∈ 𝑉) → (𝐴 ∈ (𝑁‘𝑃) ↔ ∃𝑤 ∈ 𝑈 𝐴 = (𝑤 “ {𝑃})))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∃wrex 3065  Vcvv 3429  {csn 4561   ↦ cmpt 5156  ran crn 5585   “ cima 5587  ‘cfv 6426  UnifOncust 23361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5208  ax-sep 5221  ax-nul 5228  ax-pr 5350  ax-un 7578

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3431  df-sbc 3716  df-csb 3832  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5074  df-opab 5136  df-mpt 5157  df-id 5484  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-rn 5595  df-res 5596  df-ima 5597  df-iota 6384  df-fun 6428  df-fn 6429  df-f 6430  df-f1 6431  df-fo 6432  df-f1o 6433  df-fv 6434

This theorem is referenced by:  ustuqtop1  23403  ustuqtop2  23404  ustuqtop3  23405  ustuqtop4  23406  ustuqtop5  23407  utopsnneiplem  23409