Theorem ustuqtop2 24163

Description: Lemma for ustuqtop 24167. (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 786	. . . . . . . 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 788	. . . . . . . . . 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 783	. . . . . . . . . 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 768	. . . . . . . . . 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 24128	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤 ∈ 𝑈 ∧ 𝑢 ∈ 𝑈) → (𝑤 ∩ 𝑢) ∈ 𝑈)
6	2, 3, 4, 5	syl3anc 1373	. . . . . . . . 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 4174	. . . . . . . . . . 11 ⊢ ((𝑎 = (𝑤 “ {𝑝}) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑎 ∩ 𝑏) = ((𝑤 “ {𝑝}) ∩ (𝑢 “ {𝑝})))
8		inimasn 6117	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ V → ((𝑤 ∩ 𝑢) “ {𝑝}) = ((𝑤 “ {𝑝}) ∩ (𝑢 “ {𝑝})))
9	8	elv 3449	. . . . . . . . . . 11 ⊢ ((𝑤 ∩ 𝑢) “ {𝑝}) = ((𝑤 “ {𝑝}) ∩ (𝑢 “ {𝑝}))
10	7, 9	eqtr4di 2782	. . . . . . . . . 10 ⊢ ((𝑎 = (𝑤 “ {𝑝}) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑎 ∩ 𝑏) = ((𝑤 ∩ 𝑢) “ {𝑝}))
11	10	ad4ant24 754	. . . . . . . . 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 6015	. . . . . . . . . 10 ⊢ (𝑥 = (𝑤 ∩ 𝑢) → (𝑥 “ {𝑝}) = ((𝑤 ∩ 𝑢) “ {𝑝}))
13	12	rspceeqv 3608	. . . . . . . . 9 ⊢ (((𝑤 ∩ 𝑢) ∈ 𝑈 ∧ (𝑎 ∩ 𝑏) = ((𝑤 ∩ 𝑢) “ {𝑝})) → ∃𝑥 ∈ 𝑈 (𝑎 ∩ 𝑏) = (𝑥 “ {𝑝}))
14	6, 11, 13	syl2anc 584	. . . . . . . 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 3448	. . . . . . . . . . 11 ⊢ 𝑎 ∈ V
16	15	inex1 5267	. . . . . . . . . 10 ⊢ (𝑎 ∩ 𝑏) ∈ V
17		utopustuq.1	. . . . . . . . . . 11 ⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))
18	17	ustuqtoplem 24160	. . . . . . . . . 10 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ (𝑎 ∩ 𝑏) ∈ V) → ((𝑎 ∩ 𝑏) ∈ (𝑁‘𝑝) ↔ ∃𝑥 ∈ 𝑈 (𝑎 ∩ 𝑏) = (𝑥 “ {𝑝})))
19	16, 18	mpan2 691	. . . . . . . . 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 584	. . . . . . 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 24160	. . . . . . . . . 10 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑏 ∈ V) → (𝑏 ∈ (𝑁‘𝑝) ↔ ∃𝑢 ∈ 𝑈 𝑏 = (𝑢 “ {𝑝})))
23	22	elvd 3450	. . . . . . . . 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 756	. . . . . . 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 3141	. . . . . 6 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑏 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → (𝑎 ∩ 𝑏) ∈ (𝑁‘𝑝))
27	17	ustuqtoplem 24160	. . . . . . . . 9 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ V) → (𝑎 ∈ (𝑁‘𝑝) ↔ ∃𝑤 ∈ 𝑈 𝑎 = (𝑤 “ {𝑝})))
28	27	elvd 3450	. . . . . . . 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 3141	. . . . 5 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑏 ∈ (𝑁‘𝑝)) → (𝑎 ∩ 𝑏) ∈ (𝑁‘𝑝))
32	31	ralrimiva 3125	. . . 4 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → ∀𝑏 ∈ (𝑁‘𝑝)(𝑎 ∩ 𝑏) ∈ (𝑁‘𝑝))
33	32	ralrimiva 3125	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → ∀𝑎 ∈ (𝑁‘𝑝)∀𝑏 ∈ (𝑁‘𝑝)(𝑎 ∩ 𝑏) ∈ (𝑁‘𝑝))
34		fvex 6853	. . . 4 ⊢ (𝑁‘𝑝) ∈ V
35		inficl 9352	. . . 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 218	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (fi‘(𝑁‘𝑝)) = (𝑁‘𝑝))
38		eqimss 4002	. 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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  Vcvv 3444   ∩ cin 3910   ⊆ wss 3911  {csn 4585   ↦ cmpt 5183  ran crn 5632   “ cima 5634  ‘cfv 6499  ficfi 9337  UnifOncust 24120

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-om 7823  df-1o 8411  df-2o 8412  df-en 8896  df-fin 8899  df-fi 9338  df-ust 24121

This theorem is referenced by:  ustuqtop  24167  utopsnneiplem  24168