Theorem ustuqtop1 22045

Description: Lemma for ustuqtop 22050, similar to ssnei2 20920. (Contributed by Thierry Arnoux, 11-Jan-2018.)

Hypothesis

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

Assertion

Ref	Expression
ustuqtop1	⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑏 ∈ (𝑁‘𝑝))

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

Allowed substitution hint:   𝑁(𝑣)

Proof of Theorem ustuqtop1

Dummy variables 𝑤 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1l 1112	. . . . . 6 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ (𝑎 ∈ (𝑁‘𝑝) ∧ 𝑤 ∈ 𝑈 ∧ 𝑎 = (𝑤 “ {𝑝}))) → 𝑈 ∈ (UnifOn‘𝑋))
2	1	3anassrs 1290	. . . . 5 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑈 ∈ (UnifOn‘𝑋))
3		simplr 792	. . . . 5 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑤 ∈ 𝑈)
4		ustssxp 22008	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤 ∈ 𝑈) → 𝑤 ⊆ (𝑋 × 𝑋))
5	2, 3, 4	syl2anc 693	. . . . . 6 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑤 ⊆ (𝑋 × 𝑋))
6		simpl1r 1113	. . . . . . . . 9 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ (𝑎 ∈ (𝑁‘𝑝) ∧ 𝑤 ∈ 𝑈 ∧ 𝑎 = (𝑤 “ {𝑝}))) → 𝑝 ∈ 𝑋)
7	6	3anassrs 1290	. . . . . . . 8 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑝 ∈ 𝑋)
8	7	snssd 4340	. . . . . . 7 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → {𝑝} ⊆ 𝑋)
9		simpl3 1066	. . . . . . . 8 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ (𝑎 ∈ (𝑁‘𝑝) ∧ 𝑤 ∈ 𝑈 ∧ 𝑎 = (𝑤 “ {𝑝}))) → 𝑏 ⊆ 𝑋)
10	9	3anassrs 1290	. . . . . . 7 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑏 ⊆ 𝑋)
11		xpss12 5225	. . . . . . 7 ⊢ (({𝑝} ⊆ 𝑋 ∧ 𝑏 ⊆ 𝑋) → ({𝑝} × 𝑏) ⊆ (𝑋 × 𝑋))
12	8, 10, 11	syl2anc 693	. . . . . 6 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → ({𝑝} × 𝑏) ⊆ (𝑋 × 𝑋))
13	5, 12	unssd 3789	. . . . 5 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → (𝑤 ∪ ({𝑝} × 𝑏)) ⊆ (𝑋 × 𝑋))
14		ssun1 3776	. . . . . 6 ⊢ 𝑤 ⊆ (𝑤 ∪ ({𝑝} × 𝑏))
15	14	a1i 11	. . . . 5 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑤 ⊆ (𝑤 ∪ ({𝑝} × 𝑏)))
16		ustssel 22009	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤 ∈ 𝑈 ∧ (𝑤 ∪ ({𝑝} × 𝑏)) ⊆ (𝑋 × 𝑋)) → (𝑤 ⊆ (𝑤 ∪ ({𝑝} × 𝑏)) → (𝑤 ∪ ({𝑝} × 𝑏)) ∈ 𝑈))
17	16	imp 445	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤 ∈ 𝑈 ∧ (𝑤 ∪ ({𝑝} × 𝑏)) ⊆ (𝑋 × 𝑋)) ∧ 𝑤 ⊆ (𝑤 ∪ ({𝑝} × 𝑏))) → (𝑤 ∪ ({𝑝} × 𝑏)) ∈ 𝑈)
18	2, 3, 13, 15, 17	syl31anc 1329	. . . 4 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → (𝑤 ∪ ({𝑝} × 𝑏)) ∈ 𝑈)
19		simpl2 1065	. . . . . 6 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ (𝑎 ∈ (𝑁‘𝑝) ∧ 𝑤 ∈ 𝑈 ∧ 𝑎 = (𝑤 “ {𝑝}))) → 𝑎 ⊆ 𝑏)
20	19	3anassrs 1290	. . . . 5 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑎 ⊆ 𝑏)
21		ssequn1 3783	. . . . . . 7 ⊢ (𝑎 ⊆ 𝑏 ↔ (𝑎 ∪ 𝑏) = 𝑏)
22	21	biimpi 206	. . . . . 6 ⊢ (𝑎 ⊆ 𝑏 → (𝑎 ∪ 𝑏) = 𝑏)
23		id 22	. . . . . . . 8 ⊢ (𝑎 = (𝑤 “ {𝑝}) → 𝑎 = (𝑤 “ {𝑝}))
24		inidm 3822	. . . . . . . . . . 11 ⊢ ({𝑝} ∩ {𝑝}) = {𝑝}
25		vex 3203	. . . . . . . . . . . 12 ⊢ 𝑝 ∈ V
26	25	snnz 4309	. . . . . . . . . . 11 ⊢ {𝑝} ≠ ∅
27	24, 26	eqnetri 2864	. . . . . . . . . 10 ⊢ ({𝑝} ∩ {𝑝}) ≠ ∅
28		xpima2 5578	. . . . . . . . . 10 ⊢ (({𝑝} ∩ {𝑝}) ≠ ∅ → (({𝑝} × 𝑏) “ {𝑝}) = 𝑏)
29	27, 28	mp1i 13	. . . . . . . . 9 ⊢ (𝑎 = (𝑤 “ {𝑝}) → (({𝑝} × 𝑏) “ {𝑝}) = 𝑏)
30	29	eqcomd 2628	. . . . . . . 8 ⊢ (𝑎 = (𝑤 “ {𝑝}) → 𝑏 = (({𝑝} × 𝑏) “ {𝑝}))
31	23, 30	uneq12d 3768	. . . . . . 7 ⊢ (𝑎 = (𝑤 “ {𝑝}) → (𝑎 ∪ 𝑏) = ((𝑤 “ {𝑝}) ∪ (({𝑝} × 𝑏) “ {𝑝})))
32		imaundir 5546	. . . . . . 7 ⊢ ((𝑤 ∪ ({𝑝} × 𝑏)) “ {𝑝}) = ((𝑤 “ {𝑝}) ∪ (({𝑝} × 𝑏) “ {𝑝}))
33	31, 32	syl6eqr 2674	. . . . . 6 ⊢ (𝑎 = (𝑤 “ {𝑝}) → (𝑎 ∪ 𝑏) = ((𝑤 ∪ ({𝑝} × 𝑏)) “ {𝑝}))
34	22, 33	sylan9req 2677	. . . . 5 ⊢ ((𝑎 ⊆ 𝑏 ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑏 = ((𝑤 ∪ ({𝑝} × 𝑏)) “ {𝑝}))
35	20, 34	sylancom 701	. . . 4 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑏 = ((𝑤 ∪ ({𝑝} × 𝑏)) “ {𝑝}))
36		imaeq1 5461	. . . . . 6 ⊢ (𝑢 = (𝑤 ∪ ({𝑝} × 𝑏)) → (𝑢 “ {𝑝}) = ((𝑤 ∪ ({𝑝} × 𝑏)) “ {𝑝}))
37	36	eqeq2d 2632	. . . . 5 ⊢ (𝑢 = (𝑤 ∪ ({𝑝} × 𝑏)) → (𝑏 = (𝑢 “ {𝑝}) ↔ 𝑏 = ((𝑤 ∪ ({𝑝} × 𝑏)) “ {𝑝})))
38	37	rspcev 3309	. . . 4 ⊢ (((𝑤 ∪ ({𝑝} × 𝑏)) ∈ 𝑈 ∧ 𝑏 = ((𝑤 ∪ ({𝑝} × 𝑏)) “ {𝑝})) → ∃𝑢 ∈ 𝑈 𝑏 = (𝑢 “ {𝑝}))
39	18, 35, 38	syl2anc 693	. . 3 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → ∃𝑢 ∈ 𝑈 𝑏 = (𝑢 “ {𝑝}))
40		vex 3203	. . . . . 6 ⊢ 𝑎 ∈ V
41		utopustuq.1	. . . . . . 7 ⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))
42	41	ustuqtoplem 22043	. . . . . 6 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ V) → (𝑎 ∈ (𝑁‘𝑝) ↔ ∃𝑤 ∈ 𝑈 𝑎 = (𝑤 “ {𝑝})))
43	40, 42	mpan2 707	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (𝑎 ∈ (𝑁‘𝑝) ↔ ∃𝑤 ∈ 𝑈 𝑎 = (𝑤 “ {𝑝})))
44	43	biimpa 501	. . . 4 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → ∃𝑤 ∈ 𝑈 𝑎 = (𝑤 “ {𝑝}))
45	44	3ad2antl1 1223	. . 3 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → ∃𝑤 ∈ 𝑈 𝑎 = (𝑤 “ {𝑝}))
46	39, 45	r19.29a 3078	. 2 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → ∃𝑢 ∈ 𝑈 𝑏 = (𝑢 “ {𝑝}))
47		vex 3203	. . . . 5 ⊢ 𝑏 ∈ V
48	41	ustuqtoplem 22043	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑏 ∈ V) → (𝑏 ∈ (𝑁‘𝑝) ↔ ∃𝑢 ∈ 𝑈 𝑏 = (𝑢 “ {𝑝})))
49	47, 48	mpan2 707	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (𝑏 ∈ (𝑁‘𝑝) ↔ ∃𝑢 ∈ 𝑈 𝑏 = (𝑢 “ {𝑝})))
50	49	3ad2ant1 1082	. . 3 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) → (𝑏 ∈ (𝑁‘𝑝) ↔ ∃𝑢 ∈ 𝑈 𝑏 = (𝑢 “ {𝑝})))
51	50	adantr 481	. 2 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → (𝑏 ∈ (𝑁‘𝑝) ↔ ∃𝑢 ∈ 𝑈 𝑏 = (𝑢 “ {𝑝})))
52	46, 51	mpbird 247	1 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑏 ∈ (𝑁‘𝑝))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∃wrex 2913  Vcvv 3200   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  {csn 4177   ↦ cmpt 4729   × cxp 5112  ran crn 5115   “ cima 5117  ‘cfv 5888  UnifOncust 22003

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ust 22004

This theorem is referenced by:  ustuqtop4  22048  ustuqtop  22050  utopsnneiplem  22051