Theorem smfpimbor1lem1 46818

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 45565	. 2 ⊢ (𝜑 → ∃𝑞(𝑞 ⊆ ((,) “ (ℚ × ℚ)) ∧ 𝐺 = ∪ 𝑞))
4		simprr 772	. . . . 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 46808	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ SAlg)
10	9	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ⊆ ((,) “ (ℚ × ℚ))) → 𝑇 ∈ SAlg)
11		iooex 13411	. . . . . . . . . . . 12 ⊢ (,) ∈ V
12	11	imaexi 45231	. . . . . . . . . . 11 ⊢ ((,) “ (ℚ × ℚ)) ∈ V
13	12	a1i 11	. . . . . . . . . 10 ⊢ (𝑞 ⊆ ((,) “ (ℚ × ℚ)) → ((,) “ (ℚ × ℚ)) ∈ V)
14		id 22	. . . . . . . . . 10 ⊢ (𝑞 ⊆ ((,) “ (ℚ × ℚ)) → 𝑞 ⊆ ((,) “ (ℚ × ℚ)))
15	13, 14	ssexd 5323	. . . . . . . . 9 ⊢ (𝑞 ⊆ ((,) “ (ℚ × ℚ)) → 𝑞 ∈ V)
16	15	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ⊆ ((,) “ (ℚ × ℚ))) → 𝑞 ∈ V)
17		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ⊆ ((,) “ (ℚ × ℚ))) → 𝑞 ⊆ ((,) “ (ℚ × ℚ)))
18		ioofun 45569	. . . . . . . . . . . . . . 15 ⊢ Fun (,)
19	18	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ ((,) “ (ℚ × ℚ)) → Fun (,))
20		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ ((,) “ (ℚ × ℚ)) → 𝑞 ∈ ((,) “ (ℚ × ℚ)))
21		fvelima 6973	. . . . . . . . . . . . . 14 ⊢ ((Fun (,) ∧ 𝑞 ∈ ((,) “ (ℚ × ℚ))) → ∃𝑝 ∈ (ℚ × ℚ)((,)‘𝑝) = 𝑞)
22	19, 20, 21	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ ((,) “ (ℚ × ℚ)) → ∃𝑝 ∈ (ℚ × ℚ)((,)‘𝑝) = 𝑞)
23	22	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ ((,) “ (ℚ × ℚ))) → ∃𝑝 ∈ (ℚ × ℚ)((,)‘𝑝) = 𝑞)
24		id 22	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((,)‘𝑝) = 𝑞 → ((,)‘𝑝) = 𝑞)
25	24	eqcomd 2742	. . . . . . . . . . . . . . . . . . 19 ⊢ (((,)‘𝑝) = 𝑞 → 𝑞 = ((,)‘𝑝))
26	25	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑝 ∈ (ℚ × ℚ) ∧ ((,)‘𝑝) = 𝑞) → 𝑞 = ((,)‘𝑝))
27		1st2nd2 8054	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 ∈ (ℚ × ℚ) → 𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
28	27	fveq2d 6909	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑝 ∈ (ℚ × ℚ) → ((,)‘𝑝) = ((,)‘⟨(1st ‘𝑝), (2nd ‘𝑝)⟩))
29		df-ov 7435	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1st ‘𝑝)(,)(2nd ‘𝑝)) = ((,)‘⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
30	29	eqcomi 2745	. . . . . . . . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 ∈ (ℚ × ℚ) → ((,)‘𝑝) = ((1st ‘𝑝)(,)(2nd ‘𝑝)))
33	32	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑝 ∈ (ℚ × ℚ) ∧ ((,)‘𝑝) = 𝑞) → ((,)‘𝑝) = ((1st ‘𝑝)(,)(2nd ‘𝑝)))
34	26, 33	eqtrd 2776	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑝 ∈ (ℚ × ℚ) ∧ ((,)‘𝑝) = 𝑞) → 𝑞 = ((1st ‘𝑝)(,)(2nd ‘𝑝)))
35	34	3adant1 1130	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℚ × ℚ) ∧ ((,)‘𝑝) = 𝑞) → 𝑞 = ((1st ‘𝑝)(,)(2nd ‘𝑝)))
36		ioossre 13449	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1st ‘𝑝)(,)(2nd ‘𝑝)) ⊆ ℝ
37		ovex 7465	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1st ‘𝑝)(,)(2nd ‘𝑝)) ∈ V
38	37	elpw 4603	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((1st ‘𝑝)(,)(2nd ‘𝑝)) ∈ 𝒫 ℝ ↔ ((1st ‘𝑝)(,)(2nd ‘𝑝)) ⊆ ℝ)
39	36, 38	mpbir 231	. . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℚ × ℚ)) → 𝑆 ∈ SAlg)
42	6	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℚ × ℚ)) → 𝐹 ∈ (SMblFn‘𝑆))
43		xp1st 8047	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑝 ∈ (ℚ × ℚ) → (1st ‘𝑝) ∈ ℚ)
44	43	qred 12998	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 ∈ (ℚ × ℚ) → (1st ‘𝑝) ∈ ℝ)
45	44	rexrd 11312	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 ∈ (ℚ × ℚ) → (1st ‘𝑝) ∈ ℝ*)
46	45	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℚ × ℚ)) → (1st ‘𝑝) ∈ ℝ*)
47		xp2nd 8048	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑝 ∈ (ℚ × ℚ) → (2nd ‘𝑝) ∈ ℚ)
48	47	qred 12998	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 ∈ (ℚ × ℚ) → (2nd ‘𝑝) ∈ ℝ)
49	48	rexrd 11312	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 ∈ (ℚ × ℚ) → (2nd ‘𝑝) ∈ ℝ*)
50	49	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℚ × ℚ)) → (2nd ‘𝑝) ∈ ℝ*)
51	41, 42, 7, 46, 50	smfpimioo 46807	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℚ × ℚ)) → (◡𝐹 “ ((1st ‘𝑝)(,)(2nd ‘𝑝))) ∈ (𝑆 ↾t 𝐷))
52	40, 51	jca 511	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℚ × ℚ)) → (((1st ‘𝑝)(,)(2nd ‘𝑝)) ∈ 𝒫 ℝ ∧ (◡𝐹 “ ((1st ‘𝑝)(,)(2nd ‘𝑝))) ∈ (𝑆 ↾t 𝐷)))
53		imaeq2 6073	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑒 = ((1st ‘𝑝)(,)(2nd ‘𝑝)) → (◡𝐹 “ 𝑒) = (◡𝐹 “ ((1st ‘𝑝)(,)(2nd ‘𝑝))))
54	53	eleq1d 2825	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑒 = ((1st ‘𝑝)(,)(2nd ‘𝑝)) → ((◡𝐹 “ 𝑒) ∈ (𝑆 ↾t 𝐷) ↔ (◡𝐹 “ ((1st ‘𝑝)(,)(2nd ‘𝑝))) ∈ (𝑆 ↾t 𝐷)))
55	54, 8	elrab2 3694	. . . . . . . . . . . . . . . . . 18 ⊢ (((1st ‘𝑝)(,)(2nd ‘𝑝)) ∈ 𝑇 ↔ (((1st ‘𝑝)(,)(2nd ‘𝑝)) ∈ 𝒫 ℝ ∧ (◡𝐹 “ ((1st ‘𝑝)(,)(2nd ‘𝑝))) ∈ (𝑆 ↾t 𝐷)))
56	52, 55	sylibr 234	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℚ × ℚ)) → ((1st ‘𝑝)(,)(2nd ‘𝑝)) ∈ 𝑇)
57	56	3adant3 1132	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℚ × ℚ) ∧ ((,)‘𝑝) = 𝑞) → ((1st ‘𝑝)(,)(2nd ‘𝑝)) ∈ 𝑇)
58	35, 57	eqeltrd 2840	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℚ × ℚ) ∧ ((,)‘𝑝) = 𝑞) → 𝑞 ∈ 𝑇)
59	58	3exp 1119	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑝 ∈ (ℚ × ℚ) → (((,)‘𝑝) = 𝑞 → 𝑞 ∈ 𝑇)))
60	59	rexlimdv 3152	. . . . . . . . . . . . 13 ⊢ (𝜑 → (∃𝑝 ∈ (ℚ × ℚ)((,)‘𝑝) = 𝑞 → 𝑞 ∈ 𝑇))
61	60	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ ((,) “ (ℚ × ℚ))) → (∃𝑝 ∈ (ℚ × ℚ)((,)‘𝑝) = 𝑞 → 𝑞 ∈ 𝑇))
62	23, 61	mpd 15	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ ((,) “ (ℚ × ℚ))) → 𝑞 ∈ 𝑇)
63	62	ssd 45090	. . . . . . . . . 10 ⊢ (𝜑 → ((,) “ (ℚ × ℚ)) ⊆ 𝑇)
64	63	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ⊆ ((,) “ (ℚ × ℚ))) → ((,) “ (ℚ × ℚ)) ⊆ 𝑇)
65	17, 64	sstrd 3993	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ⊆ ((,) “ (ℚ × ℚ))) → 𝑞 ⊆ 𝑇)
66	16, 65	elpwd 4605	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ⊆ ((,) “ (ℚ × ℚ))) → 𝑞 ∈ 𝒫 𝑇)
67		ssdomg 9041	. . . . . . . . . 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 45378	. . . . . . . . . . . . 13 ⊢ ℚ ≼ ω
70	69, 69	pm3.2i 470	. . . . . . . . . . . 12 ⊢ (ℚ ≼ ω ∧ ℚ ≼ ω)
71		xpct 10057	. . . . . . . . . . . 12 ⊢ ((ℚ ≼ ω ∧ ℚ ≼ ω) → (ℚ × ℚ) ≼ ω)
72	70, 71	ax-mp 5	. . . . . . . . . . 11 ⊢ (ℚ × ℚ) ≼ ω
73		fimact 10576	. . . . . . . . . . 11 ⊢ (((ℚ × ℚ) ≼ ω ∧ Fun (,)) → ((,) “ (ℚ × ℚ)) ≼ ω)
74	72, 18, 73	mp2an 692	. . . . . . . . . 10 ⊢ ((,) “ (ℚ × ℚ)) ≼ ω
75	74	a1i 11	. . . . . . . . 9 ⊢ (𝑞 ⊆ ((,) “ (ℚ × ℚ)) → ((,) “ (ℚ × ℚ)) ≼ ω)
76		domtr 9048	. . . . . . . . 9 ⊢ ((𝑞 ≼ ((,) “ (ℚ × ℚ)) ∧ ((,) “ (ℚ × ℚ)) ≼ ω) → 𝑞 ≼ ω)
77	68, 75, 76	syl2anc 584	. . . . . . . 8 ⊢ (𝑞 ⊆ ((,) “ (ℚ × ℚ)) → 𝑞 ≼ ω)
78	77	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ⊆ ((,) “ (ℚ × ℚ))) → 𝑞 ≼ ω)
79	10, 66, 78	salunicl 46336	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ⊆ ((,) “ (ℚ × ℚ))) → ∪ 𝑞 ∈ 𝑇)
80	79	adantrr 717	. . . . 5 ⊢ ((𝜑 ∧ (𝑞 ⊆ ((,) “ (ℚ × ℚ)) ∧ 𝐺 = ∪ 𝑞)) → ∪ 𝑞 ∈ 𝑇)
81	4, 80	eqeltrd 2840	. . . 4 ⊢ ((𝜑 ∧ (𝑞 ⊆ ((,) “ (ℚ × ℚ)) ∧ 𝐺 = ∪ 𝑞)) → 𝐺 ∈ 𝑇)
82	81	ex 412	. . 3 ⊢ (𝜑 → ((𝑞 ⊆ ((,) “ (ℚ × ℚ)) ∧ 𝐺 = ∪ 𝑞) → 𝐺 ∈ 𝑇))
83	82	exlimdv 1932	. 2 ⊢ (𝜑 → (∃𝑞(𝑞 ⊆ ((,) “ (ℚ × ℚ)) ∧ 𝐺 = ∪ 𝑞) → 𝐺 ∈ 𝑇))
84	3, 83	mpd 15	1 ⊢ (𝜑 → 𝐺 ∈ 𝑇)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1539  ∃wex 1778   ∈ wcel 2107  ∃wrex 3069  {crab 3435  Vcvv 3479   ⊆ wss 3950  𝒫 cpw 4599  ⟨cop 4631  ∪ cuni 4906   class class class wbr 5142   × cxp 5682  ^◡ccnv 5683  dom cdm 5684  ran crn 5685   “ cima 5687  Fun wfun 6554  ‘cfv 6560  (class class class)co 7432  ωcom 7888  1^st c1st 8013  2^nd c2nd 8014   ≼ cdom 8984  ℝcr 11155  ℝ^*cxr 11295  ℚcq 12991  (,)cioo 13388   ↾_t crest 17466  topGenctg 17483  SAlgcsalg 46328  SMblFncsmblfn 46715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682  ax-cc 10476  ax-ac2 10504  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233  ax-pre-sup 11234

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-oadd 8511  df-omul 8512  df-er 8746  df-map 8869  df-pm 8870  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-sup 9483  df-inf 9484  df-oi 9551  df-card 9980  df-acn 9983  df-ac 10157  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-div 11922  df-nn 12268  df-n0 12529  df-z 12616  df-uz 12880  df-q 12992  df-rp 13036  df-ioo 13392  df-ico 13394  df-fl 13833  df-rest 17468  df-topgen 17489  df-bases 22954  df-salg 46329  df-smblfn 46716

This theorem is referenced by:  smfpimbor1lem2  46819