Theorem smfpimbor1lem1 46769

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 45518	. 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 46759	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ SAlg)
10	9	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ⊆ ((,) “ (ℚ × ℚ))) → 𝑇 ∈ SAlg)
11		iooex 13305	. . . . . . . . . . . 12 ⊢ (,) ∈ V
12	11	imaexi 45188	. . . . . . . . . . 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 5274	. . . . . . . . 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 45522	. . . . . . . . . . . . . . 15 ⊢ Fun (,)
19	18	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ ((,) “ (ℚ × ℚ)) → Fun (,))
20		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ ((,) “ (ℚ × ℚ)) → 𝑞 ∈ ((,) “ (ℚ × ℚ)))
21		fvelima 6908	. . . . . . . . . . . . . 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 2735	. . . . . . . . . . . . . . . . . . 19 ⊢ (((,)‘𝑝) = 𝑞 → 𝑞 = ((,)‘𝑝))
26	25	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑝 ∈ (ℚ × ℚ) ∧ ((,)‘𝑝) = 𝑞) → 𝑞 = ((,)‘𝑝))
27		1st2nd2 7986	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 ∈ (ℚ × ℚ) → 𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
28	27	fveq2d 6844	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑝 ∈ (ℚ × ℚ) → ((,)‘𝑝) = ((,)‘⟨(1st ‘𝑝), (2nd ‘𝑝)⟩))
29		df-ov 7372	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1st ‘𝑝)(,)(2nd ‘𝑝)) = ((,)‘⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
30	29	eqcomi 2738	. . . . . . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . . . 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 13344	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1st ‘𝑝)(,)(2nd ‘𝑝)) ⊆ ℝ
37		ovex 7402	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1st ‘𝑝)(,)(2nd ‘𝑝)) ∈ V
38	37	elpw 4563	. . . . . . . . . . . . . . . . . . . . 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 7979	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑝 ∈ (ℚ × ℚ) → (1st ‘𝑝) ∈ ℚ)
44	43	qred 12890	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 ∈ (ℚ × ℚ) → (1st ‘𝑝) ∈ ℝ)
45	44	rexrd 11200	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 ∈ (ℚ × ℚ) → (1st ‘𝑝) ∈ ℝ*)
46	45	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℚ × ℚ)) → (1st ‘𝑝) ∈ ℝ*)
47		xp2nd 7980	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑝 ∈ (ℚ × ℚ) → (2nd ‘𝑝) ∈ ℚ)
48	47	qred 12890	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 ∈ (ℚ × ℚ) → (2nd ‘𝑝) ∈ ℝ)
49	48	rexrd 11200	. . . . . . . . . . . . . . . . . . . . 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 46758	. . . . . . . . . . . . . . . . . . 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 6016	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑒 = ((1st ‘𝑝)(,)(2nd ‘𝑝)) → (◡𝐹 “ 𝑒) = (◡𝐹 “ ((1st ‘𝑝)(,)(2nd ‘𝑝))))
54	53	eleq1d 2813	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑒 = ((1st ‘𝑝)(,)(2nd ‘𝑝)) → ((◡𝐹 “ 𝑒) ∈ (𝑆 ↾t 𝐷) ↔ (◡𝐹 “ ((1st ‘𝑝)(,)(2nd ‘𝑝))) ∈ (𝑆 ↾t 𝐷)))
55	54, 8	elrab2 3659	. . . . . . . . . . . . . . . . . 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 2828	. . . . . . . . . . . . . . 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 3132	. . . . . . . . . . . . 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 45047	. . . . . . . . . 10 ⊢ (𝜑 → ((,) “ (ℚ × ℚ)) ⊆ 𝑇)
64	63	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ⊆ ((,) “ (ℚ × ℚ))) → ((,) “ (ℚ × ℚ)) ⊆ 𝑇)
65	17, 64	sstrd 3954	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ⊆ ((,) “ (ℚ × ℚ))) → 𝑞 ⊆ 𝑇)
66	16, 65	elpwd 4565	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ⊆ ((,) “ (ℚ × ℚ))) → 𝑞 ∈ 𝒫 𝑇)
67		ssdomg 8948	. . . . . . . . . 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 45331	. . . . . . . . . . . . 13 ⊢ ℚ ≼ ω
70	69, 69	pm3.2i 470	. . . . . . . . . . . 12 ⊢ (ℚ ≼ ω ∧ ℚ ≼ ω)
71		xpct 9945	. . . . . . . . . . . 12 ⊢ ((ℚ ≼ ω ∧ ℚ ≼ ω) → (ℚ × ℚ) ≼ ω)
72	70, 71	ax-mp 5	. . . . . . . . . . 11 ⊢ (ℚ × ℚ) ≼ ω
73		fimact 10464	. . . . . . . . . . 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 8955	. . . . . . . . 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 46287	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ⊆ ((,) “ (ℚ × ℚ))) → ∪ 𝑞 ∈ 𝑇)
80	79	adantrr 717	. . . . 5 ⊢ ((𝜑 ∧ (𝑞 ⊆ ((,) “ (ℚ × ℚ)) ∧ 𝐺 = ∪ 𝑞)) → ∪ 𝑞 ∈ 𝑇)
81	4, 80	eqeltrd 2828	. . . 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 3053  {crab 3402  Vcvv 3444   ⊆ wss 3911  𝒫 cpw 4559  ⟨cop 4591  ∪ cuni 4867   class class class wbr 5102   × cxp 5629  ^◡ccnv 5630  dom cdm 5631  ran crn 5632   “ cima 5634  Fun wfun 6493  ‘cfv 6499  (class class class)co 7369  ωcom 7822  1^st c1st 7945  2^nd c2nd 7946   ≼ cdom 8893  ℝcr 11043  ℝ^*cxr 11183  ℚcq 12883  (,)cioo 13282   ↾_t crest 17359  topGenctg 17376  SAlgcsalg 46279  SMblFncsmblfn 46666

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  ax-inf2 9570  ax-cc 10364  ax-ac2 10392  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121  ax-pre-sup 11122

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-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  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-iin 4954  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-se 5585  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-pred 6262  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-isom 6508  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-2o 8412  df-oadd 8415  df-omul 8416  df-er 8648  df-map 8778  df-pm 8779  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-sup 9369  df-inf 9370  df-oi 9439  df-card 9868  df-acn 9871  df-ac 10045  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-div 11812  df-nn 12163  df-n0 12419  df-z 12506  df-uz 12770  df-q 12884  df-rp 12928  df-ioo 13286  df-ico 13288  df-fl 13730  df-rest 17361  df-topgen 17382  df-bases 22809  df-salg 46280  df-smblfn 46667

This theorem is referenced by:  smfpimbor1lem2  46770