Theorem ustuqtop1 24271

Description: Lemma for ustuqtop 24276, similar to ssnei2 23145. (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 1224	. . . . . 6 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ (𝑎 ∈ (𝑁‘𝑝) ∧ 𝑤 ∈ 𝑈 ∧ 𝑎 = (𝑤 “ {𝑝}))) → 𝑈 ∈ (UnifOn‘𝑋))
2	1	3anassrs 1360	. . . . 5 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑈 ∈ (UnifOn‘𝑋))
3		simplr 768	. . . . 5 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑤 ∈ 𝑈)
4		ustssxp 24234	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤 ∈ 𝑈) → 𝑤 ⊆ (𝑋 × 𝑋))
5	2, 3, 4	syl2anc 583	. . . . . 6 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑤 ⊆ (𝑋 × 𝑋))
6		simpl1r 1225	. . . . . . . . 9 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ (𝑎 ∈ (𝑁‘𝑝) ∧ 𝑤 ∈ 𝑈 ∧ 𝑎 = (𝑤 “ {𝑝}))) → 𝑝 ∈ 𝑋)
7	6	3anassrs 1360	. . . . . . . 8 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑝 ∈ 𝑋)
8	7	snssd 4834	. . . . . . 7 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → {𝑝} ⊆ 𝑋)
9		simpl3 1193	. . . . . . . 8 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ (𝑎 ∈ (𝑁‘𝑝) ∧ 𝑤 ∈ 𝑈 ∧ 𝑎 = (𝑤 “ {𝑝}))) → 𝑏 ⊆ 𝑋)
10	9	3anassrs 1360	. . . . . . 7 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑏 ⊆ 𝑋)
11		xpss12 5715	. . . . . . 7 ⊢ (({𝑝} ⊆ 𝑋 ∧ 𝑏 ⊆ 𝑋) → ({𝑝} × 𝑏) ⊆ (𝑋 × 𝑋))
12	8, 10, 11	syl2anc 583	. . . . . 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 4215	. . . . 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 4201	. . . . . 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 24235	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤 ∈ 𝑈 ∧ (𝑤 ∪ ({𝑝} × 𝑏)) ⊆ (𝑋 × 𝑋)) → (𝑤 ⊆ (𝑤 ∪ ({𝑝} × 𝑏)) → (𝑤 ∪ ({𝑝} × 𝑏)) ∈ 𝑈))
17	16	imp 406	. . . . 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 1373	. . . 4 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → (𝑤 ∪ ({𝑝} × 𝑏)) ∈ 𝑈)
19		simpl2 1192	. . . . . 6 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ (𝑎 ∈ (𝑁‘𝑝) ∧ 𝑤 ∈ 𝑈 ∧ 𝑎 = (𝑤 “ {𝑝}))) → 𝑎 ⊆ 𝑏)
20	19	3anassrs 1360	. . . . 5 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑎 ⊆ 𝑏)
21		ssequn1 4209	. . . . . . 7 ⊢ (𝑎 ⊆ 𝑏 ↔ (𝑎 ∪ 𝑏) = 𝑏)
22	21	biimpi 216	. . . . . 6 ⊢ (𝑎 ⊆ 𝑏 → (𝑎 ∪ 𝑏) = 𝑏)
23		id 22	. . . . . . . 8 ⊢ (𝑎 = (𝑤 “ {𝑝}) → 𝑎 = (𝑤 “ {𝑝}))
24		inidm 4248	. . . . . . . . . . 11 ⊢ ({𝑝} ∩ {𝑝}) = {𝑝}
25		vex 3492	. . . . . . . . . . . 12 ⊢ 𝑝 ∈ V
26	25	snnz 4801	. . . . . . . . . . 11 ⊢ {𝑝} ≠ ∅
27	24, 26	eqnetri 3017	. . . . . . . . . 10 ⊢ ({𝑝} ∩ {𝑝}) ≠ ∅
28		xpima2 6215	. . . . . . . . . 10 ⊢ (({𝑝} ∩ {𝑝}) ≠ ∅ → (({𝑝} × 𝑏) “ {𝑝}) = 𝑏)
29	27, 28	mp1i 13	. . . . . . . . 9 ⊢ (𝑎 = (𝑤 “ {𝑝}) → (({𝑝} × 𝑏) “ {𝑝}) = 𝑏)
30	29	eqcomd 2746	. . . . . . . 8 ⊢ (𝑎 = (𝑤 “ {𝑝}) → 𝑏 = (({𝑝} × 𝑏) “ {𝑝}))
31	23, 30	uneq12d 4192	. . . . . . 7 ⊢ (𝑎 = (𝑤 “ {𝑝}) → (𝑎 ∪ 𝑏) = ((𝑤 “ {𝑝}) ∪ (({𝑝} × 𝑏) “ {𝑝})))
32		imaundir 6182	. . . . . . 7 ⊢ ((𝑤 ∪ ({𝑝} × 𝑏)) “ {𝑝}) = ((𝑤 “ {𝑝}) ∪ (({𝑝} × 𝑏) “ {𝑝}))
33	31, 32	eqtr4di 2798	. . . . . 6 ⊢ (𝑎 = (𝑤 “ {𝑝}) → (𝑎 ∪ 𝑏) = ((𝑤 ∪ ({𝑝} × 𝑏)) “ {𝑝}))
34	22, 33	sylan9req 2801	. . . . 5 ⊢ ((𝑎 ⊆ 𝑏 ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑏 = ((𝑤 ∪ ({𝑝} × 𝑏)) “ {𝑝}))
35	20, 34	sylancom 587	. . . 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 6084	. . . . 5 ⊢ (𝑢 = (𝑤 ∪ ({𝑝} × 𝑏)) → (𝑢 “ {𝑝}) = ((𝑤 ∪ ({𝑝} × 𝑏)) “ {𝑝}))
37	36	rspceeqv 3658	. . . 4 ⊢ (((𝑤 ∪ ({𝑝} × 𝑏)) ∈ 𝑈 ∧ 𝑏 = ((𝑤 ∪ ({𝑝} × 𝑏)) “ {𝑝})) → ∃𝑢 ∈ 𝑈 𝑏 = (𝑢 “ {𝑝}))
38	18, 35, 37	syl2anc 583	. . 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 24269	. . . . . 6 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ V) → (𝑎 ∈ (𝑁‘𝑝) ↔ ∃𝑤 ∈ 𝑈 𝑎 = (𝑤 “ {𝑝})))
41	40	elvd 3494	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (𝑎 ∈ (𝑁‘𝑝) ↔ ∃𝑤 ∈ 𝑈 𝑎 = (𝑤 “ {𝑝})))
42	41	biimpa 476	. . . 4 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → ∃𝑤 ∈ 𝑈 𝑎 = (𝑤 “ {𝑝}))
43	42	3ad2antl1 1185	. . 3 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → ∃𝑤 ∈ 𝑈 𝑎 = (𝑤 “ {𝑝}))
44	38, 43	r19.29a 3168	. 2 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → ∃𝑢 ∈ 𝑈 𝑏 = (𝑢 “ {𝑝}))
45	39	ustuqtoplem 24269	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑏 ∈ V) → (𝑏 ∈ (𝑁‘𝑝) ↔ ∃𝑢 ∈ 𝑈 𝑏 = (𝑢 “ {𝑝})))
46	45	elvd 3494	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (𝑏 ∈ (𝑁‘𝑝) ↔ ∃𝑢 ∈ 𝑈 𝑏 = (𝑢 “ {𝑝})))
47	46	3ad2ant1 1133	. . 3 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) → (𝑏 ∈ (𝑁‘𝑝) ↔ ∃𝑢 ∈ 𝑈 𝑏 = (𝑢 “ {𝑝})))
48	47	adantr 480	. 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 257	1 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑏 ∈ (𝑁‘𝑝))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∃wrex 3076  Vcvv 3488   ∪ cun 3974   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  {csn 4648   ↦ cmpt 5249   × cxp 5698  ran crn 5701   “ cima 5703  ‘cfv 6573  UnifOncust 24229

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ust 24230

This theorem is referenced by:  ustuqtop4  24274  ustuqtop  24276  utopsnneiplem  24277