Theorem ustuqtop5 24472

Description: Lemma for ustuqtop 24473. (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 24434	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ∈ 𝑈)
2		snssi 4746	. . . . . . . . 9 ⊢ (𝑝 ∈ 𝑋 → {𝑝} ⊆ 𝑋)
3		dfss 3918	. . . . . . . . 9 ⊢ ({𝑝} ⊆ 𝑋 ↔ {𝑝} = ({𝑝} ∩ 𝑋))
4	2, 3	sylib 221	. . . . . . . 8 ⊢ (𝑝 ∈ 𝑋 → {𝑝} = ({𝑝} ∩ 𝑋))
5		incom 4155	. . . . . . . 8 ⊢ ({𝑝} ∩ 𝑋) = (𝑋 ∩ {𝑝})
6	4, 5	eqtr2di 2812	. . . . . . 7 ⊢ (𝑝 ∈ 𝑋 → (𝑋 ∩ {𝑝}) = {𝑝})
7		snnzg 4735	. . . . . . 7 ⊢ (𝑝 ∈ 𝑋 → {𝑝} ≠ ∅)
8	6, 7	eqnetrd 3022	. . . . . 6 ⊢ (𝑝 ∈ 𝑋 → (𝑋 ∩ {𝑝}) ≠ ∅)
9	8	adantl 487	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (𝑋 ∩ {𝑝}) ≠ ∅)
10		xpima2 6177	. . . . 5 ⊢ ((𝑋 ∩ {𝑝}) ≠ ∅ → ((𝑋 × 𝑋) “ {𝑝}) = 𝑋)
11	9, 10	syl 18	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → ((𝑋 × 𝑋) “ {𝑝}) = 𝑋)
12	11	eqcomd 2766	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → 𝑋 = ((𝑋 × 𝑋) “ {𝑝}))
13		imaeq1 6051	. . . 4 ⊢ (𝑤 = (𝑋 × 𝑋) → (𝑤 “ {𝑝}) = ((𝑋 × 𝑋) “ {𝑝}))
14	13	rspceeqv 3599	. . 3 ⊢ (((𝑋 × 𝑋) ∈ 𝑈 ∧ 𝑋 = ((𝑋 × 𝑋) “ {𝑝})) → ∃𝑤 ∈ 𝑈 𝑋 = (𝑤 “ {𝑝}))
15	1, 12, 14	syl2an2r 698	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → ∃𝑤 ∈ 𝑈 𝑋 = (𝑤 “ {𝑝}))
16		elfvex 6914	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
17		utopustuq.1	. . . 4 ⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))
18	17	ustuqtoplem 24466	. . 3 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑋 ∈ V) → (𝑋 ∈ (𝑁‘𝑝) ↔ ∃𝑤 ∈ 𝑈 𝑋 = (𝑤 “ {𝑝})))
19	16, 18	mpidan 702	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (𝑋 ∈ (𝑁‘𝑝) ↔ ∃𝑤 ∈ 𝑈 𝑋 = (𝑤 “ {𝑝})))
20	15, 19	mpbird 260	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → 𝑋 ∈ (𝑁‘𝑝))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∃wrex 3086  Vcvv 3450   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  {csn 4584   ↦ cmpt 5186   × cxp 5653  ran crn 5656   “ cima 5658  ‘cfv 6533  UnifOncust 24427

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ust 24428

This theorem is used by:  ustuqtop  24473  utopsnneiplem  24474