Theorem ustuqtop1 23139

Description: Lemma for ustuqtop 23144, similar to ssnei2 22013. (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 1226	. . . . . 6 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ (𝑎 ∈ (𝑁‘𝑝) ∧ 𝑤 ∈ 𝑈 ∧ 𝑎 = (𝑤 “ {𝑝}))) → 𝑈 ∈ (UnifOn‘𝑋))
2	1	3anassrs 1362	. . . . 5 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑈 ∈ (UnifOn‘𝑋))
3		simplr 769	. . . . 5 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑤 ∈ 𝑈)
4		ustssxp 23102	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤 ∈ 𝑈) → 𝑤 ⊆ (𝑋 × 𝑋))
5	2, 3, 4	syl2anc 587	. . . . . 6 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑤 ⊆ (𝑋 × 𝑋))
6		simpl1r 1227	. . . . . . . . 9 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ (𝑎 ∈ (𝑁‘𝑝) ∧ 𝑤 ∈ 𝑈 ∧ 𝑎 = (𝑤 “ {𝑝}))) → 𝑝 ∈ 𝑋)
7	6	3anassrs 1362	. . . . . . . 8 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑝 ∈ 𝑋)
8	7	snssd 4722	. . . . . . 7 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → {𝑝} ⊆ 𝑋)
9		simpl3 1195	. . . . . . . 8 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ (𝑎 ∈ (𝑁‘𝑝) ∧ 𝑤 ∈ 𝑈 ∧ 𝑎 = (𝑤 “ {𝑝}))) → 𝑏 ⊆ 𝑋)
10	9	3anassrs 1362	. . . . . . 7 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑏 ⊆ 𝑋)
11		xpss12 5566	. . . . . . 7 ⊢ (({𝑝} ⊆ 𝑋 ∧ 𝑏 ⊆ 𝑋) → ({𝑝} × 𝑏) ⊆ (𝑋 × 𝑋))
12	8, 10, 11	syl2anc 587	. . . . . 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 4100	. . . . 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 4086	. . . . . 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 23103	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤 ∈ 𝑈 ∧ (𝑤 ∪ ({𝑝} × 𝑏)) ⊆ (𝑋 × 𝑋)) → (𝑤 ⊆ (𝑤 ∪ ({𝑝} × 𝑏)) → (𝑤 ∪ ({𝑝} × 𝑏)) ∈ 𝑈))
17	16	imp 410	. . . . 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 1375	. . . 4 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → (𝑤 ∪ ({𝑝} × 𝑏)) ∈ 𝑈)
19		simpl2 1194	. . . . . 6 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ (𝑎 ∈ (𝑁‘𝑝) ∧ 𝑤 ∈ 𝑈 ∧ 𝑎 = (𝑤 “ {𝑝}))) → 𝑎 ⊆ 𝑏)
20	19	3anassrs 1362	. . . . 5 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑎 ⊆ 𝑏)
21		ssequn1 4094	. . . . . . 7 ⊢ (𝑎 ⊆ 𝑏 ↔ (𝑎 ∪ 𝑏) = 𝑏)
22	21	biimpi 219	. . . . . 6 ⊢ (𝑎 ⊆ 𝑏 → (𝑎 ∪ 𝑏) = 𝑏)
23		id 22	. . . . . . . 8 ⊢ (𝑎 = (𝑤 “ {𝑝}) → 𝑎 = (𝑤 “ {𝑝}))
24		inidm 4133	. . . . . . . . . . 11 ⊢ ({𝑝} ∩ {𝑝}) = {𝑝}
25		vex 3412	. . . . . . . . . . . 12 ⊢ 𝑝 ∈ V
26	25	snnz 4692	. . . . . . . . . . 11 ⊢ {𝑝} ≠ ∅
27	24, 26	eqnetri 3011	. . . . . . . . . 10 ⊢ ({𝑝} ∩ {𝑝}) ≠ ∅
28		xpima2 6047	. . . . . . . . . 10 ⊢ (({𝑝} ∩ {𝑝}) ≠ ∅ → (({𝑝} × 𝑏) “ {𝑝}) = 𝑏)
29	27, 28	mp1i 13	. . . . . . . . 9 ⊢ (𝑎 = (𝑤 “ {𝑝}) → (({𝑝} × 𝑏) “ {𝑝}) = 𝑏)
30	29	eqcomd 2743	. . . . . . . 8 ⊢ (𝑎 = (𝑤 “ {𝑝}) → 𝑏 = (({𝑝} × 𝑏) “ {𝑝}))
31	23, 30	uneq12d 4078	. . . . . . 7 ⊢ (𝑎 = (𝑤 “ {𝑝}) → (𝑎 ∪ 𝑏) = ((𝑤 “ {𝑝}) ∪ (({𝑝} × 𝑏) “ {𝑝})))
32		imaundir 6014	. . . . . . 7 ⊢ ((𝑤 ∪ ({𝑝} × 𝑏)) “ {𝑝}) = ((𝑤 “ {𝑝}) ∪ (({𝑝} × 𝑏) “ {𝑝}))
33	31, 32	eqtr4di 2796	. . . . . 6 ⊢ (𝑎 = (𝑤 “ {𝑝}) → (𝑎 ∪ 𝑏) = ((𝑤 ∪ ({𝑝} × 𝑏)) “ {𝑝}))
34	22, 33	sylan9req 2799	. . . . 5 ⊢ ((𝑎 ⊆ 𝑏 ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑏 = ((𝑤 ∪ ({𝑝} × 𝑏)) “ {𝑝}))
35	20, 34	sylancom 591	. . . 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 5924	. . . . 5 ⊢ (𝑢 = (𝑤 ∪ ({𝑝} × 𝑏)) → (𝑢 “ {𝑝}) = ((𝑤 ∪ ({𝑝} × 𝑏)) “ {𝑝}))
37	36	rspceeqv 3552	. . . 4 ⊢ (((𝑤 ∪ ({𝑝} × 𝑏)) ∈ 𝑈 ∧ 𝑏 = ((𝑤 ∪ ({𝑝} × 𝑏)) “ {𝑝})) → ∃𝑢 ∈ 𝑈 𝑏 = (𝑢 “ {𝑝}))
38	18, 35, 37	syl2anc 587	. . 3 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → ∃𝑢 ∈ 𝑈 𝑏 = (𝑢 “ {𝑝}))
39		utopustuq.1	. . . . . . 7 ⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))
40	39	ustuqtoplem 23137	. . . . . 6 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ V) → (𝑎 ∈ (𝑁‘𝑝) ↔ ∃𝑤 ∈ 𝑈 𝑎 = (𝑤 “ {𝑝})))
41	40	elvd 3415	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (𝑎 ∈ (𝑁‘𝑝) ↔ ∃𝑤 ∈ 𝑈 𝑎 = (𝑤 “ {𝑝})))
42	41	biimpa 480	. . . 4 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → ∃𝑤 ∈ 𝑈 𝑎 = (𝑤 “ {𝑝}))
43	42	3ad2antl1 1187	. . 3 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → ∃𝑤 ∈ 𝑈 𝑎 = (𝑤 “ {𝑝}))
44	38, 43	r19.29a 3208	. 2 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → ∃𝑢 ∈ 𝑈 𝑏 = (𝑢 “ {𝑝}))
45	39	ustuqtoplem 23137	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑏 ∈ V) → (𝑏 ∈ (𝑁‘𝑝) ↔ ∃𝑢 ∈ 𝑈 𝑏 = (𝑢 “ {𝑝})))
46	45	elvd 3415	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (𝑏 ∈ (𝑁‘𝑝) ↔ ∃𝑢 ∈ 𝑈 𝑏 = (𝑢 “ {𝑝})))
47	46	3ad2ant1 1135	. . 3 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) → (𝑏 ∈ (𝑁‘𝑝) ↔ ∃𝑢 ∈ 𝑈 𝑏 = (𝑢 “ {𝑝})))
48	47	adantr 484	. 2 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → (𝑏 ∈ (𝑁‘𝑝) ↔ ∃𝑢 ∈ 𝑈 𝑏 = (𝑢 “ {𝑝})))
49	44, 48	mpbird 260	1 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑏 ∈ (𝑁‘𝑝))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110   ≠ wne 2940  ∃wrex 3062  Vcvv 3408   ∪ cun 3864   ∩ cin 3865   ⊆ wss 3866  ∅c0 4237  {csn 4541   ↦ cmpt 5135   × cxp 5549  ran crn 5552   “ cima 5554  ‘cfv 6380  UnifOncust 23097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ust 23098

This theorem is referenced by:  ustuqtop4  23142  ustuqtop  23144  utopsnneiplem  23145