Theorem ustuqtop2 23302

Description: Lemma for ustuqtop 23306. (Contributed by Thierry Arnoux, 11-Jan-2018.)

Hypothesis

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

Assertion

Ref	Expression
ustuqtop2	⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (fi‘(𝑁‘𝑝)) ⊆ (𝑁‘𝑝))

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

Allowed substitution hint:   𝑁(𝑣)

Proof of Theorem ustuqtop2

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

Step	Hyp	Ref	Expression
1		simp-6l 783	. . . . . . . 8 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑏 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢 ∈ 𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋))
2		simp-7l 785	. . . . . . . . . 10 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑏 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢 ∈ 𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → 𝑈 ∈ (UnifOn‘𝑋))
3		simp-4r 780	. . . . . . . . . 10 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑏 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢 ∈ 𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → 𝑤 ∈ 𝑈)
4		simplr 765	. . . . . . . . . 10 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑏 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢 ∈ 𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → 𝑢 ∈ 𝑈)
5		ustincl 23267	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤 ∈ 𝑈 ∧ 𝑢 ∈ 𝑈) → (𝑤 ∩ 𝑢) ∈ 𝑈)
6	2, 3, 4, 5	syl3anc 1369	. . . . . . . . 9 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑏 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢 ∈ 𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑤 ∩ 𝑢) ∈ 𝑈)
7		ineq12 4138	. . . . . . . . . . 11 ⊢ ((𝑎 = (𝑤 “ {𝑝}) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑎 ∩ 𝑏) = ((𝑤 “ {𝑝}) ∩ (𝑢 “ {𝑝})))
8		inimasn 6048	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ V → ((𝑤 ∩ 𝑢) “ {𝑝}) = ((𝑤 “ {𝑝}) ∩ (𝑢 “ {𝑝})))
9	8	elv 3428	. . . . . . . . . . 11 ⊢ ((𝑤 ∩ 𝑢) “ {𝑝}) = ((𝑤 “ {𝑝}) ∩ (𝑢 “ {𝑝}))
10	7, 9	eqtr4di 2797	. . . . . . . . . 10 ⊢ ((𝑎 = (𝑤 “ {𝑝}) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑎 ∩ 𝑏) = ((𝑤 ∩ 𝑢) “ {𝑝}))
11	10	ad4ant24 750	. . . . . . . . 9 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑏 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢 ∈ 𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑎 ∩ 𝑏) = ((𝑤 ∩ 𝑢) “ {𝑝}))
12		imaeq1 5953	. . . . . . . . . 10 ⊢ (𝑥 = (𝑤 ∩ 𝑢) → (𝑥 “ {𝑝}) = ((𝑤 ∩ 𝑢) “ {𝑝}))
13	12	rspceeqv 3567	. . . . . . . . 9 ⊢ (((𝑤 ∩ 𝑢) ∈ 𝑈 ∧ (𝑎 ∩ 𝑏) = ((𝑤 ∩ 𝑢) “ {𝑝})) → ∃𝑥 ∈ 𝑈 (𝑎 ∩ 𝑏) = (𝑥 “ {𝑝}))
14	6, 11, 13	syl2anc 583	. . . . . . . 8 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑏 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢 ∈ 𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → ∃𝑥 ∈ 𝑈 (𝑎 ∩ 𝑏) = (𝑥 “ {𝑝}))
15		vex 3426	. . . . . . . . . . 11 ⊢ 𝑎 ∈ V
16	15	inex1 5236	. . . . . . . . . 10 ⊢ (𝑎 ∩ 𝑏) ∈ V
17		utopustuq.1	. . . . . . . . . . 11 ⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))
18	17	ustuqtoplem 23299	. . . . . . . . . 10 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ (𝑎 ∩ 𝑏) ∈ V) → ((𝑎 ∩ 𝑏) ∈ (𝑁‘𝑝) ↔ ∃𝑥 ∈ 𝑈 (𝑎 ∩ 𝑏) = (𝑥 “ {𝑝})))
19	16, 18	mpan2 687	. . . . . . . . 9 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → ((𝑎 ∩ 𝑏) ∈ (𝑁‘𝑝) ↔ ∃𝑥 ∈ 𝑈 (𝑎 ∩ 𝑏) = (𝑥 “ {𝑝})))
20	19	biimpar 477	. . . . . . . 8 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ ∃𝑥 ∈ 𝑈 (𝑎 ∩ 𝑏) = (𝑥 “ {𝑝})) → (𝑎 ∩ 𝑏) ∈ (𝑁‘𝑝))
21	1, 14, 20	syl2anc 583	. . . . . . 7 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑏 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢 ∈ 𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑎 ∩ 𝑏) ∈ (𝑁‘𝑝))
22	17	ustuqtoplem 23299	. . . . . . . . . 10 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑏 ∈ V) → (𝑏 ∈ (𝑁‘𝑝) ↔ ∃𝑢 ∈ 𝑈 𝑏 = (𝑢 “ {𝑝})))
23	22	elvd 3429	. . . . . . . . 9 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (𝑏 ∈ (𝑁‘𝑝) ↔ ∃𝑢 ∈ 𝑈 𝑏 = (𝑢 “ {𝑝})))
24	23	biimpa 476	. . . . . . . 8 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑏 ∈ (𝑁‘𝑝)) → ∃𝑢 ∈ 𝑈 𝑏 = (𝑢 “ {𝑝}))
25	24	ad5ant13 753	. . . . . . 7 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑏 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → ∃𝑢 ∈ 𝑈 𝑏 = (𝑢 “ {𝑝}))
26	21, 25	r19.29a 3217	. . . . . 6 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑏 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → (𝑎 ∩ 𝑏) ∈ (𝑁‘𝑝))
27	17	ustuqtoplem 23299	. . . . . . . . 9 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ V) → (𝑎 ∈ (𝑁‘𝑝) ↔ ∃𝑤 ∈ 𝑈 𝑎 = (𝑤 “ {𝑝})))
28	27	elvd 3429	. . . . . . . 8 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (𝑎 ∈ (𝑁‘𝑝) ↔ ∃𝑤 ∈ 𝑈 𝑎 = (𝑤 “ {𝑝})))
29	28	biimpa 476	. . . . . . 7 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → ∃𝑤 ∈ 𝑈 𝑎 = (𝑤 “ {𝑝}))
30	29	adantr 480	. . . . . 6 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑏 ∈ (𝑁‘𝑝)) → ∃𝑤 ∈ 𝑈 𝑎 = (𝑤 “ {𝑝}))
31	26, 30	r19.29a 3217	. . . . 5 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑏 ∈ (𝑁‘𝑝)) → (𝑎 ∩ 𝑏) ∈ (𝑁‘𝑝))
32	31	ralrimiva 3107	. . . 4 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → ∀𝑏 ∈ (𝑁‘𝑝)(𝑎 ∩ 𝑏) ∈ (𝑁‘𝑝))
33	32	ralrimiva 3107	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → ∀𝑎 ∈ (𝑁‘𝑝)∀𝑏 ∈ (𝑁‘𝑝)(𝑎 ∩ 𝑏) ∈ (𝑁‘𝑝))
34		fvex 6769	. . . 4 ⊢ (𝑁‘𝑝) ∈ V
35		inficl 9114	. . . 4 ⊢ ((𝑁‘𝑝) ∈ V → (∀𝑎 ∈ (𝑁‘𝑝)∀𝑏 ∈ (𝑁‘𝑝)(𝑎 ∩ 𝑏) ∈ (𝑁‘𝑝) ↔ (fi‘(𝑁‘𝑝)) = (𝑁‘𝑝)))
36	34, 35	ax-mp 5	. . 3 ⊢ (∀𝑎 ∈ (𝑁‘𝑝)∀𝑏 ∈ (𝑁‘𝑝)(𝑎 ∩ 𝑏) ∈ (𝑁‘𝑝) ↔ (fi‘(𝑁‘𝑝)) = (𝑁‘𝑝))
37	33, 36	sylib 217	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (fi‘(𝑁‘𝑝)) = (𝑁‘𝑝))
38		eqimss 3973	. 2 ⊢ ((fi‘(𝑁‘𝑝)) = (𝑁‘𝑝) → (fi‘(𝑁‘𝑝)) ⊆ (𝑁‘𝑝))
39	37, 38	syl 17	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (fi‘(𝑁‘𝑝)) ⊆ (𝑁‘𝑝))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ∩ cin 3882   ⊆ wss 3883  {csn 4558   ↦ cmpt 5153  ran crn 5581   “ cima 5583  ‘cfv 6418  ficfi 9099  UnifOncust 23259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-om 7688  df-1o 8267  df-er 8456  df-en 8692  df-fin 8695  df-fi 9100  df-ust 23260

This theorem is referenced by:  ustuqtop  23306  utopsnneiplem  23307