Theorem ustuqtoplem 24181

Description: Lemma for ustuqtop 24188. (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 482	. . . . . . . . . 10 ⊢ ((𝑝 = 𝑞 ∧ 𝑣 ∈ 𝑈) → 𝑝 = 𝑞)
3	2	sneqd 4590	. . . . . . . . 9 ⊢ ((𝑝 = 𝑞 ∧ 𝑣 ∈ 𝑈) → {𝑝} = {𝑞})
4	3	imaeq2d 6017	. . . . . . . 8 ⊢ ((𝑝 = 𝑞 ∧ 𝑣 ∈ 𝑈) → (𝑣 “ {𝑝}) = (𝑣 “ {𝑞}))
5	4	mpteq2dva 5189	. . . . . . 7 ⊢ (𝑝 = 𝑞 → (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) = (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))
6	5	rneqd 5885	. . . . . 6 ⊢ (𝑝 = 𝑞 → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))
7	6	cbvmptv 5200	. . . . 5 ⊢ (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝}))) = (𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))
8	1, 7	eqtri 2757	. . . 4 ⊢ 𝑁 = (𝑞 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})))
9		simpr2 1196	. . . . . . . . 9 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑃 ∈ 𝑋 ∧ 𝑞 = 𝑃 ∧ 𝑣 ∈ 𝑈)) → 𝑞 = 𝑃)
10	9	sneqd 4590	. . . . . . . 8 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑃 ∈ 𝑋 ∧ 𝑞 = 𝑃 ∧ 𝑣 ∈ 𝑈)) → {𝑞} = {𝑃})
11	10	imaeq2d 6017	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑃 ∈ 𝑋 ∧ 𝑞 = 𝑃 ∧ 𝑣 ∈ 𝑈)) → (𝑣 “ {𝑞}) = (𝑣 “ {𝑃}))
12	11	3anassrs 1361	. . . . . 6 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑞 = 𝑃) ∧ 𝑣 ∈ 𝑈) → (𝑣 “ {𝑞}) = (𝑣 “ {𝑃}))
13	12	mpteq2dva 5189	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑞 = 𝑃) → (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})) = (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))
14	13	rneqd 5885	. . . 4 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑞 = 𝑃) → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑞})) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))
15		simpr 484	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → 𝑃 ∈ 𝑋)
16		mptexg 7165	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
17		rnexg 7842	. . . . . 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 480	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ∈ V)
20	8, 14, 15, 19	fvmptd2 6947	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → (𝑁‘𝑃) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))
21	20	eleq2d 2820	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → (𝐴 ∈ (𝑁‘𝑃) ↔ 𝐴 ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃}))))
22		imaeq1 6012	. . . 4 ⊢ (𝑣 = 𝑤 → (𝑣 “ {𝑃}) = (𝑤 “ {𝑃}))
23	22	cbvmptv 5200	. . 3 ⊢ (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) = (𝑤 ∈ 𝑈 ↦ (𝑤 “ {𝑃}))
24	23	elrnmpt 5905	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ↔ ∃𝑤 ∈ 𝑈 𝐴 = (𝑤 “ {𝑃})))
25	21, 24	sylan9bb 509	1 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝐴 ∈ 𝑉) → (𝐴 ∈ (𝑁‘𝑃) ↔ ∃𝑤 ∈ 𝑈 𝐴 = (𝑤 “ {𝑃})))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∃wrex 3058  Vcvv 3438  {csn 4578   ↦ cmpt 5177  ran crn 5623   “ cima 5625  ‘cfv 6490  UnifOncust 24142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498

This theorem is referenced by:  ustuqtop1  24183  ustuqtop2  24184  ustuqtop3  24185  ustuqtop4  24186  ustuqtop5  24187  utopsnneiplem  24189