Theorem ustuqtop2 24588

Description: Lemma for ustuqtop 24592. (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 799	. . . . . . . 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 801	. . . . . . . . . 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 796	. . . . . . . . . 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 781	. . . . . . . . . 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 24554	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤 ∈ 𝑈 ∧ 𝑢 ∈ 𝑈) → (𝑤 ∩ 𝑢) ∈ 𝑈)
6	2, 3, 4, 5	syl3anc 1399	. . . . . . . . 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 4115	. . . . . . . . . . 11 ⊢ ((𝑎 = (𝑤 “ {𝑝}) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑎 ∩ 𝑏) = ((𝑤 “ {𝑝}) ∩ (𝑢 “ {𝑝})))
8		inimasn 6157	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ V → ((𝑤 ∩ 𝑢) “ {𝑝}) = ((𝑤 “ {𝑝}) ∩ (𝑢 “ {𝑝})))
9	8	elv 3459	. . . . . . . . . . 11 ⊢ ((𝑤 ∩ 𝑢) “ {𝑝}) = ((𝑤 “ {𝑝}) ∩ (𝑢 “ {𝑝}))
10	7, 9	eqtr4di 2816	. . . . . . . . . 10 ⊢ ((𝑎 = (𝑤 “ {𝑝}) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑎 ∩ 𝑏) = ((𝑤 ∩ 𝑢) “ {𝑝}))
11	10	ad4ant24 767	. . . . . . . . 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 6058	. . . . . . . . . 10 ⊢ (𝑥 = (𝑤 ∩ 𝑢) → (𝑥 “ {𝑝}) = ((𝑤 ∩ 𝑢) “ {𝑝}))
13	12	rspceeqv 3602	. . . . . . . . 9 ⊢ (((𝑤 ∩ 𝑢) ∈ 𝑈 ∧ (𝑎 ∩ 𝑏) = ((𝑤 ∩ 𝑢) “ {𝑝})) → ∃𝑥 ∈ 𝑈 (𝑎 ∩ 𝑏) = (𝑥 “ {𝑝}))
14	6, 11, 13	syl2anc 596	. . . . . . . 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 3458	. . . . . . . . . . 11 ⊢ 𝑎 ∈ V
16	15	inex1 5281	. . . . . . . . . 10 ⊢ (𝑎 ∩ 𝑏) ∈ V
17		utopustuq.1	. . . . . . . . . . 11 ⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))
18	17	ustuqtoplem 24585	. . . . . . . . . 10 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ (𝑎 ∩ 𝑏) ∈ V) → ((𝑎 ∩ 𝑏) ∈ (𝑁‘𝑝) ↔ ∃𝑥 ∈ 𝑈 (𝑎 ∩ 𝑏) = (𝑥 “ {𝑝})))
19	16, 18	mpan2 704	. . . . . . . . 9 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → ((𝑎 ∩ 𝑏) ∈ (𝑁‘𝑝) ↔ ∃𝑥 ∈ 𝑈 (𝑎 ∩ 𝑏) = (𝑥 “ {𝑝})))
20	19	biimpar 483	. . . . . . . 8 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ ∃𝑥 ∈ 𝑈 (𝑎 ∩ 𝑏) = (𝑥 “ {𝑝})) → (𝑎 ∩ 𝑏) ∈ (𝑁‘𝑝))
21	1, 14, 20	syl2anc 596	. . . . . . 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 24585	. . . . . . . . . 10 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑏 ∈ V) → (𝑏 ∈ (𝑁‘𝑝) ↔ ∃𝑢 ∈ 𝑈 𝑏 = (𝑢 “ {𝑝})))
23	22	elvd 3460	. . . . . . . . 9 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (𝑏 ∈ (𝑁‘𝑝) ↔ ∃𝑢 ∈ 𝑈 𝑏 = (𝑢 “ {𝑝})))
24	23	biimpa 482	. . . . . . . 8 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑏 ∈ (𝑁‘𝑝)) → ∃𝑢 ∈ 𝑈 𝑏 = (𝑢 “ {𝑝}))
25	24	ad5ant13 769	. . . . . . 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 3174	. . . . . 6 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑏 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → (𝑎 ∩ 𝑏) ∈ (𝑁‘𝑝))
27	17	ustuqtoplem 24585	. . . . . . . . 9 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ V) → (𝑎 ∈ (𝑁‘𝑝) ↔ ∃𝑤 ∈ 𝑈 𝑎 = (𝑤 “ {𝑝})))
28	27	elvd 3460	. . . . . . . 8 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (𝑎 ∈ (𝑁‘𝑝) ↔ ∃𝑤 ∈ 𝑈 𝑎 = (𝑤 “ {𝑝})))
29	28	biimpa 482	. . . . . . 7 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → ∃𝑤 ∈ 𝑈 𝑎 = (𝑤 “ {𝑝}))
30	29	adantr 486	. . . . . 6 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑏 ∈ (𝑁‘𝑝)) → ∃𝑤 ∈ 𝑈 𝑎 = (𝑤 “ {𝑝}))
31	26, 30	r19.29a 3174	. . . . 5 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑏 ∈ (𝑁‘𝑝)) → (𝑎 ∩ 𝑏) ∈ (𝑁‘𝑝))
32	31	ralrimiva 3158	. . . 4 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → ∀𝑏 ∈ (𝑁‘𝑝)(𝑎 ∩ 𝑏) ∈ (𝑁‘𝑝))
33	32	ralrimiva 3158	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → ∀𝑎 ∈ (𝑁‘𝑝)∀𝑏 ∈ (𝑁‘𝑝)(𝑎 ∩ 𝑏) ∈ (𝑁‘𝑝))
34		fvex 6918	. . . 4 ⊢ (𝑁‘𝑝) ∈ V
35		inficl 9441	. . . 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 221	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (fi‘(𝑁‘𝑝)) = (𝑁‘𝑝))
38		eqimss 3971	. 2 ⊢ ((fi‘(𝑁‘𝑝)) = (𝑁‘𝑝) → (fi‘(𝑁‘𝑝)) ⊆ (𝑁‘𝑝))
39	37, 38	syl 18	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (fi‘(𝑁‘𝑝)) ⊆ (𝑁‘𝑝))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092  {csn 4588   ↦ cmpt 5189  ran crn 5656   “ cima 5658  ‘cfv 6555  ficfi 9426  UnifOncust 24546

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-om 7899  df-1o 8501  df-2o 8502  df-en 8999  df-fin 9002  df-fi 9427  df-ust 24547

This theorem is used by:  ustuqtop  24592  utopsnneiplem  24593