Theorem smfpimbor1lem1 47777

Description: Every open set belongs to 𝑇. This is the second step in the proof of Proposition 121E (f) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
smfpimbor1lem1.s	⊢ (𝜑 → 𝑆 ∈ SAlg)
smfpimbor1lem1.f	⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆))
smfpimbor1lem1.a	⊢ 𝐷 = dom 𝐹
smfpimbor1lem1.j	⊢ 𝐽 = (topGen‘ran (,))
smfpimbor1lem1.8	⊢ (𝜑 → 𝐺 ∈ 𝐽)
smfpimbor1lem1.t	⊢ 𝑇 = {𝑒 ∈ 𝒫 ℝ ∣ (◡𝐹 “ 𝑒) ∈ (𝑆 ↾t 𝐷)}

Assertion

Ref	Expression
smfpimbor1lem1	⊢ (𝜑 → 𝐺 ∈ 𝑇)

Distinct variable groups:   𝐷,𝑒   𝑒,𝐹   𝑆,𝑒   𝜑,𝑒

Allowed substitution hints:   𝑇(𝑒)   𝐺(𝑒)   𝐽(𝑒)

Proof of Theorem smfpimbor1lem1

Dummy variables 𝑞 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		smfpimbor1lem1.j	. . 3 ⊢ 𝐽 = (topGen‘ran (,))
2		smfpimbor1lem1.8	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐽)
3	1, 2	tgqioo2 46528	. 2 ⊢ (𝜑 → ∃𝑞(𝑞 ⊆ ((,) “ (ℚ × ℚ)) ∧ 𝐺 = ∪ 𝑞))
4		simprr 785	. . . . 5 ⊢ ((𝜑 ∧ (𝑞 ⊆ ((,) “ (ℚ × ℚ)) ∧ 𝐺 = ∪ 𝑞)) → 𝐺 = ∪ 𝑞)
5		smfpimbor1lem1.s	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ SAlg)
6		smfpimbor1lem1.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆))
7		smfpimbor1lem1.a	. . . . . . . . 9 ⊢ 𝐷 = dom 𝐹
8		smfpimbor1lem1.t	. . . . . . . . 9 ⊢ 𝑇 = {𝑒 ∈ 𝒫 ℝ ∣ (◡𝐹 “ 𝑒) ∈ (𝑆 ↾t 𝐷)}
9	5, 6, 7, 8	smfresal 47767	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ SAlg)
10	9	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ⊆ ((,) “ (ℚ × ℚ))) → 𝑇 ∈ SAlg)
11		iooex 13492	. . . . . . . . . . . 12 ⊢ (,) ∈ V
12	11	imaexi 46203	. . . . . . . . . . 11 ⊢ ((,) “ (ℚ × ℚ)) ∈ V
13	12	a1i 11	. . . . . . . . . 10 ⊢ (𝑞 ⊆ ((,) “ (ℚ × ℚ)) → ((,) “ (ℚ × ℚ)) ∈ V)
14		id 23	. . . . . . . . . 10 ⊢ (𝑞 ⊆ ((,) “ (ℚ × ℚ)) → 𝑞 ⊆ ((,) “ (ℚ × ℚ)))
15	13, 14	ssexd 5286	. . . . . . . . 9 ⊢ (𝑞 ⊆ ((,) “ (ℚ × ℚ)) → 𝑞 ∈ V)
16	15	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ⊆ ((,) “ (ℚ × ℚ))) → 𝑞 ∈ V)
17		simpr 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ⊆ ((,) “ (ℚ × ℚ))) → 𝑞 ⊆ ((,) “ (ℚ × ℚ)))
18		ioofun 46532	. . . . . . . . . . . . . . 15 ⊢ Fun (,)
19	18	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ ((,) “ (ℚ × ℚ)) → Fun (,))
20		id 23	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ ((,) “ (ℚ × ℚ)) → 𝑞 ∈ ((,) “ (ℚ × ℚ)))
21		fvelima 6948	. . . . . . . . . . . . . 14 ⊢ ((Fun (,) ∧ 𝑞 ∈ ((,) “ (ℚ × ℚ))) → ∃𝑝 ∈ (ℚ × ℚ)((,)‘𝑝) = 𝑞)
22	19, 20, 21	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ ((,) “ (ℚ × ℚ)) → ∃𝑝 ∈ (ℚ × ℚ)((,)‘𝑝) = 𝑞)
23	22	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ ((,) “ (ℚ × ℚ))) → ∃𝑝 ∈ (ℚ × ℚ)((,)‘𝑝) = 𝑞)
24		id 23	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((,)‘𝑝) = 𝑞 → ((,)‘𝑝) = 𝑞)
25	24	eqcomd 2767	. . . . . . . . . . . . . . . . . . 19 ⊢ (((,)‘𝑝) = 𝑞 → 𝑞 = ((,)‘𝑝))
26	25	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑝 ∈ (ℚ × ℚ) ∧ ((,)‘𝑝) = 𝑞) → 𝑞 = ((,)‘𝑝))
27		1st2nd2 8038	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 ∈ (ℚ × ℚ) → 𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
28	27	fveq2d 6887	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑝 ∈ (ℚ × ℚ) → ((,)‘𝑝) = ((,)‘⟨(1st ‘𝑝), (2nd ‘𝑝)⟩))
29		df-ov 7421	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1st ‘𝑝)(,)(2nd ‘𝑝)) = ((,)‘⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
30	29	eqcomi 2770	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((,)‘⟨(1st ‘𝑝), (2nd ‘𝑝)⟩) = ((1st ‘𝑝)(,)(2nd ‘𝑝))
31	30	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑝 ∈ (ℚ × ℚ) → ((,)‘⟨(1st ‘𝑝), (2nd ‘𝑝)⟩) = ((1st ‘𝑝)(,)(2nd ‘𝑝)))
32	28, 31	eqtrd 2796	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 ∈ (ℚ × ℚ) → ((,)‘𝑝) = ((1st ‘𝑝)(,)(2nd ‘𝑝)))
33	32	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑝 ∈ (ℚ × ℚ) ∧ ((,)‘𝑝) = 𝑞) → ((,)‘𝑝) = ((1st ‘𝑝)(,)(2nd ‘𝑝)))
34	26, 33	eqtrd 2796	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑝 ∈ (ℚ × ℚ) ∧ ((,)‘𝑝) = 𝑞) → 𝑞 = ((1st ‘𝑝)(,)(2nd ‘𝑝)))
35	34	3adant1 1148	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℚ × ℚ) ∧ ((,)‘𝑝) = 𝑞) → 𝑞 = ((1st ‘𝑝)(,)(2nd ‘𝑝)))
36		ioossre 13531	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1st ‘𝑝)(,)(2nd ‘𝑝)) ⊆ ℝ
37		ovex 7451	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1st ‘𝑝)(,)(2nd ‘𝑝)) ∈ V
38	37	elpw 4561	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((1st ‘𝑝)(,)(2nd ‘𝑝)) ∈ 𝒫 ℝ ↔ ((1st ‘𝑝)(,)(2nd ‘𝑝)) ⊆ ℝ)
39	36, 38	mpbir 234	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1st ‘𝑝)(,)(2nd ‘𝑝)) ∈ 𝒫 ℝ
40	39	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℚ × ℚ)) → ((1st ‘𝑝)(,)(2nd ‘𝑝)) ∈ 𝒫 ℝ)
41	5	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℚ × ℚ)) → 𝑆 ∈ SAlg)
42	6	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℚ × ℚ)) → 𝐹 ∈ (SMblFn‘𝑆))
43		xp1st 8031	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑝 ∈ (ℚ × ℚ) → (1st ‘𝑝) ∈ ℚ)
44	43	qred 13075	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 ∈ (ℚ × ℚ) → (1st ‘𝑝) ∈ ℝ)
45	44	rexrd 11352	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 ∈ (ℚ × ℚ) → (1st ‘𝑝) ∈ ℝ*)
46	45	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℚ × ℚ)) → (1st ‘𝑝) ∈ ℝ*)
47		xp2nd 8032	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑝 ∈ (ℚ × ℚ) → (2nd ‘𝑝) ∈ ℚ)
48	47	qred 13075	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 ∈ (ℚ × ℚ) → (2nd ‘𝑝) ∈ ℝ)
49	48	rexrd 11352	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 ∈ (ℚ × ℚ) → (2nd ‘𝑝) ∈ ℝ*)
50	49	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℚ × ℚ)) → (2nd ‘𝑝) ∈ ℝ*)
51	41, 42, 7, 46, 50	smfpimioo 47766	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℚ × ℚ)) → (◡𝐹 “ ((1st ‘𝑝)(,)(2nd ‘𝑝))) ∈ (𝑆 ↾t 𝐷))
52	40, 51	jca 521	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℚ × ℚ)) → (((1st ‘𝑝)(,)(2nd ‘𝑝)) ∈ 𝒫 ℝ ∧ (◡𝐹 “ ((1st ‘𝑝)(,)(2nd ‘𝑝))) ∈ (𝑆 ↾t 𝐷)))
53		imaeq2 6048	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑒 = ((1st ‘𝑝)(,)(2nd ‘𝑝)) → (◡𝐹 “ 𝑒) = (◡𝐹 “ ((1st ‘𝑝)(,)(2nd ‘𝑝))))
54	53	eleq1d 2846	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑒 = ((1st ‘𝑝)(,)(2nd ‘𝑝)) → ((◡𝐹 “ 𝑒) ∈ (𝑆 ↾t 𝐷) ↔ (◡𝐹 “ ((1st ‘𝑝)(,)(2nd ‘𝑝))) ∈ (𝑆 ↾t 𝐷)))
55	54, 8	elrab2 3649	. . . . . . . . . . . . . . . . . 18 ⊢ (((1st ‘𝑝)(,)(2nd ‘𝑝)) ∈ 𝑇 ↔ (((1st ‘𝑝)(,)(2nd ‘𝑝)) ∈ 𝒫 ℝ ∧ (◡𝐹 “ ((1st ‘𝑝)(,)(2nd ‘𝑝))) ∈ (𝑆 ↾t 𝐷)))
56	52, 55	sylibr 237	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℚ × ℚ)) → ((1st ‘𝑝)(,)(2nd ‘𝑝)) ∈ 𝑇)
57	56	3adant3 1150	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℚ × ℚ) ∧ ((,)‘𝑝) = 𝑞) → ((1st ‘𝑝)(,)(2nd ‘𝑝)) ∈ 𝑇)
58	35, 57	eqeltrd 2861	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℚ × ℚ) ∧ ((,)‘𝑝) = 𝑞) → 𝑞 ∈ 𝑇)
59	58	3exp 1137	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑝 ∈ (ℚ × ℚ) → (((,)‘𝑝) = 𝑞 → 𝑞 ∈ 𝑇)))
60	59	rexlimdv 3162	. . . . . . . . . . . . 13 ⊢ (𝜑 → (∃𝑝 ∈ (ℚ × ℚ)((,)‘𝑝) = 𝑞 → 𝑞 ∈ 𝑇))
61	60	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ ((,) “ (ℚ × ℚ))) → (∃𝑝 ∈ (ℚ × ℚ)((,)‘𝑝) = 𝑞 → 𝑞 ∈ 𝑇))
62	23, 61	mpd 16	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ ((,) “ (ℚ × ℚ))) → 𝑞 ∈ 𝑇)
63	62	ssd 46066	. . . . . . . . . 10 ⊢ (𝜑 → ((,) “ (ℚ × ℚ)) ⊆ 𝑇)
64	63	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ⊆ ((,) “ (ℚ × ℚ))) → ((,) “ (ℚ × ℚ)) ⊆ 𝑇)
65	17, 64	sstrd 3941	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ⊆ ((,) “ (ℚ × ℚ))) → 𝑞 ⊆ 𝑇)
66	16, 65	elpwd 4563	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ⊆ ((,) “ (ℚ × ℚ))) → 𝑞 ∈ 𝒫 𝑇)
67		ssdomg 9020	. . . . . . . . . 10 ⊢ (((,) “ (ℚ × ℚ)) ∈ V → (𝑞 ⊆ ((,) “ (ℚ × ℚ)) → 𝑞 ≼ ((,) “ (ℚ × ℚ))))
68	12, 67	ax-mp 5	. . . . . . . . 9 ⊢ (𝑞 ⊆ ((,) “ (ℚ × ℚ)) → 𝑞 ≼ ((,) “ (ℚ × ℚ)))
69		qct 46343	. . . . . . . . . . . . 13 ⊢ ℚ ≼ ω
70	69, 69	pm3.2i 476	. . . . . . . . . . . 12 ⊢ (ℚ ≼ ω ∧ ℚ ≼ ω)
71		xpct 10088	. . . . . . . . . . . 12 ⊢ ((ℚ ≼ ω ∧ ℚ ≼ ω) → (ℚ × ℚ) ≼ ω)
72	70, 71	ax-mp 5	. . . . . . . . . . 11 ⊢ (ℚ × ℚ) ≼ ω
73		fimact 10608	. . . . . . . . . . 11 ⊢ (((ℚ × ℚ) ≼ ω ∧ Fun (,)) → ((,) “ (ℚ × ℚ)) ≼ ω)
74	72, 18, 73	mp2an 705	. . . . . . . . . 10 ⊢ ((,) “ (ℚ × ℚ)) ≼ ω
75	74	a1i 11	. . . . . . . . 9 ⊢ (𝑞 ⊆ ((,) “ (ℚ × ℚ)) → ((,) “ (ℚ × ℚ)) ≼ ω)
76		domtr 9027	. . . . . . . . 9 ⊢ ((𝑞 ≼ ((,) “ (ℚ × ℚ)) ∧ ((,) “ (ℚ × ℚ)) ≼ ω) → 𝑞 ≼ ω)
77	68, 75, 76	syl2anc 596	. . . . . . . 8 ⊢ (𝑞 ⊆ ((,) “ (ℚ × ℚ)) → 𝑞 ≼ ω)
78	77	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ⊆ ((,) “ (ℚ × ℚ))) → 𝑞 ≼ ω)
79	10, 66, 78	salunicl 47295	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ⊆ ((,) “ (ℚ × ℚ))) → ∪ 𝑞 ∈ 𝑇)
80	79	adantrr 730	. . . . 5 ⊢ ((𝜑 ∧ (𝑞 ⊆ ((,) “ (ℚ × ℚ)) ∧ 𝐺 = ∪ 𝑞)) → ∪ 𝑞 ∈ 𝑇)
81	4, 80	eqeltrd 2861	. . . 4 ⊢ ((𝜑 ∧ (𝑞 ⊆ ((,) “ (ℚ × ℚ)) ∧ 𝐺 = ∪ 𝑞)) → 𝐺 ∈ 𝑇)
82	81	ex 418	. . 3 ⊢ (𝜑 → ((𝑞 ⊆ ((,) “ (ℚ × ℚ)) ∧ 𝐺 = ∪ 𝑞) → 𝐺 ∈ 𝑇))
83	82	exlimdv 1966	. 2 ⊢ (𝜑 → (∃𝑞(𝑞 ⊆ ((,) “ (ℚ × ℚ)) ∧ 𝐺 = ∪ 𝑞) → 𝐺 ∈ 𝑇))
84	3, 83	mpd 16	1 ⊢ (𝜑 → 𝐺 ∈ 𝑇)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃wrex 3087  {crab 3413  Vcvv 3451   ⊆ wss 3899  𝒫 cpw 4557  ⟨cop 4590  ∪ cuni 4867   class class class wbr 5103   × cxp 5649  ^◡ccnv 5650  dom cdm 5651  ran crn 5652   “ cima 5654  Fun wfun 6531  ‘cfv 6537  (class class class)co 7418  ωcom 7875  1^st c1st 7997  2^nd c2nd 7998   ≼ cdom 8964  ℝcr 11192  ℝ^*cxr 11335  ℚcq 13068  (,)cioo 13469   ↾_t crest 17584  topGenctg 17601  SAlgcsalg 47287  SMblFncsmblfn 47674

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-inf2 9635  ax-cc 10506  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270  ax-pre-sup 11271

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-2o 8470  df-oadd 8473  df-omul 8474  df-er 8710  df-map 8842  df-pm 8843  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-sup 9427  df-inf 9428  df-oi 9497  df-card 10013  df-acn 10016  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-div 11967  df-nn 12329  df-n0 12600  df-z 12687  df-uz 12959  df-q 13069  df-rp 13114  df-ioo 13473  df-ico 13475  df-fl 13925  df-rest 17586  df-topgen 17607  df-bases 23257  df-salg 47288  df-smblfn 47675

This theorem is used by:  smfpimbor1lem2  47778