Theorem smfpimbor1lem1 46794

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 45543	. 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 46784	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ SAlg)
10	9	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ⊆ ((,) “ (ℚ × ℚ))) → 𝑇 ∈ SAlg)
11		iooex 13390	. . . . . . . . . . . 12 ⊢ (,) ∈ V
12	11	imaexi 45212	. . . . . . . . . . 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 5299	. . . . . . . . 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 45547	. . . . . . . . . . . . . . 15 ⊢ Fun (,)
19	18	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ ((,) “ (ℚ × ℚ)) → Fun (,))
20		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ ((,) “ (ℚ × ℚ)) → 𝑞 ∈ ((,) “ (ℚ × ℚ)))
21		fvelima 6949	. . . . . . . . . . . . . 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 8032	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 ∈ (ℚ × ℚ) → 𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
28	27	fveq2d 6885	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑝 ∈ (ℚ × ℚ) → ((,)‘𝑝) = ((,)‘⟨(1st ‘𝑝), (2nd ‘𝑝)⟩))
29		df-ov 7413	. . . . . . . . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . . . 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 13429	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1st ‘𝑝)(,)(2nd ‘𝑝)) ⊆ ℝ
37		ovex 7443	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1st ‘𝑝)(,)(2nd ‘𝑝)) ∈ V
38	37	elpw 4584	. . . . . . . . . . . . . . . . . . . . 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 8025	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑝 ∈ (ℚ × ℚ) → (1st ‘𝑝) ∈ ℚ)
44	43	qred 12976	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 ∈ (ℚ × ℚ) → (1st ‘𝑝) ∈ ℝ)
45	44	rexrd 11290	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 ∈ (ℚ × ℚ) → (1st ‘𝑝) ∈ ℝ*)
46	45	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℚ × ℚ)) → (1st ‘𝑝) ∈ ℝ*)
47		xp2nd 8026	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑝 ∈ (ℚ × ℚ) → (2nd ‘𝑝) ∈ ℚ)
48	47	qred 12976	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 ∈ (ℚ × ℚ) → (2nd ‘𝑝) ∈ ℝ)
49	48	rexrd 11290	. . . . . . . . . . . . . . . . . . . . 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 46783	. . . . . . . . . . . . . . . . . . 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 6048	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑒 = ((1st ‘𝑝)(,)(2nd ‘𝑝)) → (◡𝐹 “ 𝑒) = (◡𝐹 “ ((1st ‘𝑝)(,)(2nd ‘𝑝))))
54	53	eleq1d 2820	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑒 = ((1st ‘𝑝)(,)(2nd ‘𝑝)) → ((◡𝐹 “ 𝑒) ∈ (𝑆 ↾t 𝐷) ↔ (◡𝐹 “ ((1st ‘𝑝)(,)(2nd ‘𝑝))) ∈ (𝑆 ↾t 𝐷)))
55	54, 8	elrab2 3679	. . . . . . . . . . . . . . . . . 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 2835	. . . . . . . . . . . . . . 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 3140	. . . . . . . . . . . . 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 45071	. . . . . . . . . 10 ⊢ (𝜑 → ((,) “ (ℚ × ℚ)) ⊆ 𝑇)
64	63	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ⊆ ((,) “ (ℚ × ℚ))) → ((,) “ (ℚ × ℚ)) ⊆ 𝑇)
65	17, 64	sstrd 3974	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ⊆ ((,) “ (ℚ × ℚ))) → 𝑞 ⊆ 𝑇)
66	16, 65	elpwd 4586	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ⊆ ((,) “ (ℚ × ℚ))) → 𝑞 ∈ 𝒫 𝑇)
67		ssdomg 9019	. . . . . . . . . 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 45356	. . . . . . . . . . . . 13 ⊢ ℚ ≼ ω
70	69, 69	pm3.2i 470	. . . . . . . . . . . 12 ⊢ (ℚ ≼ ω ∧ ℚ ≼ ω)
71		xpct 10035	. . . . . . . . . . . 12 ⊢ ((ℚ ≼ ω ∧ ℚ ≼ ω) → (ℚ × ℚ) ≼ ω)
72	70, 71	ax-mp 5	. . . . . . . . . . 11 ⊢ (ℚ × ℚ) ≼ ω
73		fimact 10554	. . . . . . . . . . 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 9026	. . . . . . . . 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 46312	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ⊆ ((,) “ (ℚ × ℚ))) → ∪ 𝑞 ∈ 𝑇)
80	79	adantrr 717	. . . . 5 ⊢ ((𝜑 ∧ (𝑞 ⊆ ((,) “ (ℚ × ℚ)) ∧ 𝐺 = ∪ 𝑞)) → ∪ 𝑞 ∈ 𝑇)
81	4, 80	eqeltrd 2835	. . . 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 1933	. 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 1540  ∃wex 1779   ∈ wcel 2109  ∃wrex 3061  {crab 3420  Vcvv 3464   ⊆ wss 3931  𝒫 cpw 4580  ⟨cop 4612  ∪ cuni 4888   class class class wbr 5124   × cxp 5657  ^◡ccnv 5658  dom cdm 5659  ran crn 5660   “ cima 5662  Fun wfun 6530  ‘cfv 6536  (class class class)co 7410  ωcom 7866  1^st c1st 7991  2^nd c2nd 7992   ≼ cdom 8962  ℝcr 11133  ℝ^*cxr 11273  ℚcq 12969  (,)cioo 13367   ↾_t crest 17439  topGenctg 17456  SAlgcsalg 46304  SMblFncsmblfn 46691

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 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-inf2 9660  ax-cc 10454  ax-ac2 10482  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211  ax-pre-sup 11212

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-iin 4975  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-oadd 8489  df-omul 8490  df-er 8724  df-map 8847  df-pm 8848  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9459  df-inf 9460  df-oi 9529  df-card 9958  df-acn 9961  df-ac 10135  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-div 11900  df-nn 12246  df-n0 12507  df-z 12594  df-uz 12858  df-q 12970  df-rp 13014  df-ioo 13371  df-ico 13373  df-fl 13814  df-rest 17441  df-topgen 17462  df-bases 22889  df-salg 46305  df-smblfn 46692

This theorem is referenced by:  smfpimbor1lem2  46795