Theorem ustuqtop5 22171

Description: Lemma for ustuqtop 22172. (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 22132	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ∈ 𝑈)
2	1	adantr 472	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (𝑋 × 𝑋) ∈ 𝑈)
3		snssi 4447	. . . . . . . . 9 ⊢ (𝑝 ∈ 𝑋 → {𝑝} ⊆ 𝑋)
4		dfss 3695	. . . . . . . . 9 ⊢ ({𝑝} ⊆ 𝑋 ↔ {𝑝} = ({𝑝} ∩ 𝑋))
5	3, 4	sylib 208	. . . . . . . 8 ⊢ (𝑝 ∈ 𝑋 → {𝑝} = ({𝑝} ∩ 𝑋))
6		incom 3913	. . . . . . . 8 ⊢ ({𝑝} ∩ 𝑋) = (𝑋 ∩ {𝑝})
7	5, 6	syl6req 2775	. . . . . . 7 ⊢ (𝑝 ∈ 𝑋 → (𝑋 ∩ {𝑝}) = {𝑝})
8		snnzg 4414	. . . . . . 7 ⊢ (𝑝 ∈ 𝑋 → {𝑝} ≠ ∅)
9	7, 8	eqnetrd 2963	. . . . . 6 ⊢ (𝑝 ∈ 𝑋 → (𝑋 ∩ {𝑝}) ≠ ∅)
10	9	adantl 473	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (𝑋 ∩ {𝑝}) ≠ ∅)
11		xpima2 5688	. . . . 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 2730	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → 𝑋 = ((𝑋 × 𝑋) “ {𝑝}))
14		imaeq1 5571	. . . . 5 ⊢ (𝑤 = (𝑋 × 𝑋) → (𝑤 “ {𝑝}) = ((𝑋 × 𝑋) “ {𝑝}))
15	14	eqeq2d 2734	. . . 4 ⊢ (𝑤 = (𝑋 × 𝑋) → (𝑋 = (𝑤 “ {𝑝}) ↔ 𝑋 = ((𝑋 × 𝑋) “ {𝑝})))
16	15	rspcev 3413	. . 3 ⊢ (((𝑋 × 𝑋) ∈ 𝑈 ∧ 𝑋 = ((𝑋 × 𝑋) “ {𝑝})) → ∃𝑤 ∈ 𝑈 𝑋 = (𝑤 “ {𝑝}))
17	2, 13, 16	syl2anc 696	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → ∃𝑤 ∈ 𝑈 𝑋 = (𝑤 “ {𝑝}))
18		elfvex 6334	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
19		utopustuq.1	. . . 4 ⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))
20	19	ustuqtoplem 22165	. . 3 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑋 ∈ V) → (𝑋 ∈ (𝑁‘𝑝) ↔ ∃𝑤 ∈ 𝑈 𝑋 = (𝑤 “ {𝑝})))
21	18, 20	mpidan 707	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (𝑋 ∈ (𝑁‘𝑝) ↔ ∃𝑤 ∈ 𝑈 𝑋 = (𝑤 “ {𝑝})))
22	17, 21	mpbird 247	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → 𝑋 ∈ (𝑁‘𝑝))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1596   ∈ wcel 2103   ≠ wne 2896  ∃wrex 3015  Vcvv 3304   ∩ cin 3679   ⊆ wss 3680  ∅c0 4023  {csn 4285   ↦ cmpt 4837   × cxp 5216  ran crn 5219   “ cima 5221  ‘cfv 6001  UnifOncust 22125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-ust 22126

This theorem is referenced by:  ustuqtop  22172  utopsnneiplem  22173