Theorem ustuqtop5 21859

Description: Lemma for ustuqtop 21860. (Contributed by Thierry Arnoux, 11-Jan-2018.)

Hypothesis

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

Assertion

Ref	Expression
ustuqtop5	⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → 𝑋 ∈ (𝑁‘𝑝))

Distinct variable groups:   𝑣,𝑝,𝑈   𝑋,𝑝,𝑣   𝑁,𝑝

Allowed substitution hint:   𝑁(𝑣)

Proof of Theorem ustuqtop5

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ustbasel 21820	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ∈ 𝑈)
2	1	adantr 480	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (𝑋 × 𝑋) ∈ 𝑈)
3		snssi 4280	. . . . . . . . 9 ⊢ (𝑝 ∈ 𝑋 → {𝑝} ⊆ 𝑋)
4		dfss 3555	. . . . . . . . 9 ⊢ ({𝑝} ⊆ 𝑋 ↔ {𝑝} = ({𝑝} ∩ 𝑋))
5	3, 4	sylib 207	. . . . . . . 8 ⊢ (𝑝 ∈ 𝑋 → {𝑝} = ({𝑝} ∩ 𝑋))
6		incom 3767	. . . . . . . 8 ⊢ ({𝑝} ∩ 𝑋) = (𝑋 ∩ {𝑝})
7	5, 6	syl6req 2661	. . . . . . 7 ⊢ (𝑝 ∈ 𝑋 → (𝑋 ∩ {𝑝}) = {𝑝})
8		snnzg 4251	. . . . . . 7 ⊢ (𝑝 ∈ 𝑋 → {𝑝} ≠ ∅)
9	7, 8	eqnetrd 2849	. . . . . 6 ⊢ (𝑝 ∈ 𝑋 → (𝑋 ∩ {𝑝}) ≠ ∅)
10	9	adantl 481	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (𝑋 ∩ {𝑝}) ≠ ∅)
11		xpima2 5497	. . . . 5 ⊢ ((𝑋 ∩ {𝑝}) ≠ ∅ → ((𝑋 × 𝑋) “ {𝑝}) = 𝑋)
12	10, 11	syl 17	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → ((𝑋 × 𝑋) “ {𝑝}) = 𝑋)
13	12	eqcomd 2616	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → 𝑋 = ((𝑋 × 𝑋) “ {𝑝}))
14		imaeq1 5380	. . . . 5 ⊢ (𝑤 = (𝑋 × 𝑋) → (𝑤 “ {𝑝}) = ((𝑋 × 𝑋) “ {𝑝}))
15	14	eqeq2d 2620	. . . 4 ⊢ (𝑤 = (𝑋 × 𝑋) → (𝑋 = (𝑤 “ {𝑝}) ↔ 𝑋 = ((𝑋 × 𝑋) “ {𝑝})))
16	15	rspcev 3282	. . 3 ⊢ (((𝑋 × 𝑋) ∈ 𝑈 ∧ 𝑋 = ((𝑋 × 𝑋) “ {𝑝})) → ∃𝑤 ∈ 𝑈 𝑋 = (𝑤 “ {𝑝}))
17	2, 13, 16	syl2anc 691	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → ∃𝑤 ∈ 𝑈 𝑋 = (𝑤 “ {𝑝}))
18		elfvex 6131	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
19		utopustuq.1	. . . 4 ⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))
20	19	ustuqtoplem 21853	. . 3 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑋 ∈ V) → (𝑋 ∈ (𝑁‘𝑝) ↔ ∃𝑤 ∈ 𝑈 𝑋 = (𝑤 “ {𝑝})))
21	18, 20	mpidan 701	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (𝑋 ∈ (𝑁‘𝑝) ↔ ∃𝑤 ∈ 𝑈 𝑋 = (𝑤 “ {𝑝})))
22	17, 21	mpbird 246	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → 𝑋 ∈ (𝑁‘𝑝))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {csn 4125   ↦ cmpt 4643   × cxp 5036  ran crn 5039   “ cima 5041  ‘cfv 5804  UnifOncust 21813

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ust 21814

This theorem is referenced by:  ustuqtop  21860  utopsnneiplem  21861