smfpimbor1lem1 - Mathbox for Glauco Siliprandi

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Theorem smfpimbor1lem1 45999

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 44745	. 2 ⊢ (𝜑 → ∃𝑞(𝑞 ⊆ ((,) “ (ℚ × ℚ)) ∧ 𝐺 = ∪ 𝑞))
4		simprr 770	. . . . 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 45989	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ SAlg)
10	9	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ⊆ ((,) “ (ℚ × ℚ))) → 𝑇 ∈ SAlg)
11		iooex 13344	. . . . . . . . . . . 12 ⊢ (,) ∈ V
12	11	imaexi 44405	. . . . . . . . . . 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 5314	. . . . . . . . 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 44749	. . . . . . . . . . . . . . 15 ⊢ Fun (,)
19	18	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ ((,) “ (ℚ × ℚ)) → Fun (,))
20		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ ((,) “ (ℚ × ℚ)) → 𝑞 ∈ ((,) “ (ℚ × ℚ)))
21		fvelima 6947	. . . . . . . . . . . . . 14 ⊢ ((Fun (,) ∧ 𝑞 ∈ ((,) “ (ℚ × ℚ))) → ∃𝑝 ∈ (ℚ × ℚ)((,)‘𝑝) = 𝑞)
22	19, 20, 21	syl2anc 583	. . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . . . . . . 19 ⊢ (((,)‘𝑝) = 𝑞 → 𝑞 = ((,)‘𝑝))
26	25	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑝 ∈ (ℚ × ℚ) ∧ ((,)‘𝑝) = 𝑞) → 𝑞 = ((,)‘𝑝))
27		1st2nd2 8007	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 ∈ (ℚ × ℚ) → 𝑝 = ⟨(1^st ‘𝑝), (2^nd ‘𝑝)⟩)
28	27	fveq2d 6885	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑝 ∈ (ℚ × ℚ) → ((,)‘𝑝) = ((,)‘⟨(1^st ‘𝑝), (2^nd ‘𝑝)⟩))
29		df-ov 7404	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1^st ‘𝑝)(,)(2^nd ‘𝑝)) = ((,)‘⟨(1^st ‘𝑝), (2^nd ‘𝑝)⟩)
30	29	eqcomi 2733	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((,)‘⟨(1^st ‘𝑝), (2^nd ‘𝑝)⟩) = ((1^st ‘𝑝)(,)(2^nd ‘𝑝))
31	30	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑝 ∈ (ℚ × ℚ) → ((,)‘⟨(1^st ‘𝑝), (2^nd ‘𝑝)⟩) = ((1^st ‘𝑝)(,)(2^nd ‘𝑝)))
32	28, 31	eqtrd 2764	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 ∈ (ℚ × ℚ) → ((,)‘𝑝) = ((1^st ‘𝑝)(,)(2^nd ‘𝑝)))
33	32	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑝 ∈ (ℚ × ℚ) ∧ ((,)‘𝑝) = 𝑞) → ((,)‘𝑝) = ((1^st ‘𝑝)(,)(2^nd ‘𝑝)))
34	26, 33	eqtrd 2764	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑝 ∈ (ℚ × ℚ) ∧ ((,)‘𝑝) = 𝑞) → 𝑞 = ((1^st ‘𝑝)(,)(2^nd ‘𝑝)))
35	34	3adant1 1127	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℚ × ℚ) ∧ ((,)‘𝑝) = 𝑞) → 𝑞 = ((1^st ‘𝑝)(,)(2^nd ‘𝑝)))
36		ioossre 13382	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1^st ‘𝑝)(,)(2^nd ‘𝑝)) ⊆ ℝ
37		ovex 7434	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1^st ‘𝑝)(,)(2^nd ‘𝑝)) ∈ V
38	37	elpw 4598	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((1^st ‘𝑝)(,)(2^nd ‘𝑝)) ∈ 𝒫 ℝ ↔ ((1^st ‘𝑝)(,)(2^nd ‘𝑝)) ⊆ ℝ)
39	36, 38	mpbir 230	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1^st ‘𝑝)(,)(2^nd ‘𝑝)) ∈ 𝒫 ℝ
40	39	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℚ × ℚ)) → ((1^st ‘𝑝)(,)(2^nd ‘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 8000	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑝 ∈ (ℚ × ℚ) → (1^st ‘𝑝) ∈ ℚ)
44	43	qred 12936	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 ∈ (ℚ × ℚ) → (1^st ‘𝑝) ∈ ℝ)
45	44	rexrd 11261	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 ∈ (ℚ × ℚ) → (1^st ‘𝑝) ∈ ℝ^*)
46	45	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℚ × ℚ)) → (1^st ‘𝑝) ∈ ℝ^*)
47		xp2nd 8001	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑝 ∈ (ℚ × ℚ) → (2^nd ‘𝑝) ∈ ℚ)
48	47	qred 12936	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 ∈ (ℚ × ℚ) → (2^nd ‘𝑝) ∈ ℝ)
49	48	rexrd 11261	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 ∈ (ℚ × ℚ) → (2^nd ‘𝑝) ∈ ℝ^*)
50	49	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℚ × ℚ)) → (2^nd ‘𝑝) ∈ ℝ^*)
51	41, 42, 7, 46, 50	smfpimioo 45988	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℚ × ℚ)) → (^◡𝐹 “ ((1^st ‘𝑝)(,)(2^nd ‘𝑝))) ∈ (𝑆 ↾_t 𝐷))
52	40, 51	jca 511	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℚ × ℚ)) → (((1^st ‘𝑝)(,)(2^nd ‘𝑝)) ∈ 𝒫 ℝ ∧ (^◡𝐹 “ ((1^st ‘𝑝)(,)(2^nd ‘𝑝))) ∈ (𝑆 ↾_t 𝐷)))
53		imaeq2 6045	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑒 = ((1^st ‘𝑝)(,)(2^nd ‘𝑝)) → (^◡𝐹 “ 𝑒) = (^◡𝐹 “ ((1^st ‘𝑝)(,)(2^nd ‘𝑝))))
54	53	eleq1d 2810	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑒 = ((1^st ‘𝑝)(,)(2^nd ‘𝑝)) → ((^◡𝐹 “ 𝑒) ∈ (𝑆 ↾_t 𝐷) ↔ (^◡𝐹 “ ((1^st ‘𝑝)(,)(2^nd ‘𝑝))) ∈ (𝑆 ↾_t 𝐷)))
55	54, 8	elrab2 3678	. . . . . . . . . . . . . . . . . 18 ⊢ (((1^st ‘𝑝)(,)(2^nd ‘𝑝)) ∈ 𝑇 ↔ (((1^st ‘𝑝)(,)(2^nd ‘𝑝)) ∈ 𝒫 ℝ ∧ (^◡𝐹 “ ((1^st ‘𝑝)(,)(2^nd ‘𝑝))) ∈ (𝑆 ↾_t 𝐷)))
56	52, 55	sylibr 233	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℚ × ℚ)) → ((1^st ‘𝑝)(,)(2^nd ‘𝑝)) ∈ 𝑇)
57	56	3adant3 1129	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℚ × ℚ) ∧ ((,)‘𝑝) = 𝑞) → ((1^st ‘𝑝)(,)(2^nd ‘𝑝)) ∈ 𝑇)
58	35, 57	eqeltrd 2825	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℚ × ℚ) ∧ ((,)‘𝑝) = 𝑞) → 𝑞 ∈ 𝑇)
59	58	3exp 1116	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑝 ∈ (ℚ × ℚ) → (((,)‘𝑝) = 𝑞 → 𝑞 ∈ 𝑇)))
60	59	rexlimdv 3145	. . . . . . . . . . . . 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 44257	. . . . . . . . . 10 ⊢ (𝜑 → ((,) “ (ℚ × ℚ)) ⊆ 𝑇)
64	63	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ⊆ ((,) “ (ℚ × ℚ))) → ((,) “ (ℚ × ℚ)) ⊆ 𝑇)
65	17, 64	sstrd 3984	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ⊆ ((,) “ (ℚ × ℚ))) → 𝑞 ⊆ 𝑇)
66	16, 65	elpwd 4600	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ⊆ ((,) “ (ℚ × ℚ))) → 𝑞 ∈ 𝒫 𝑇)
67		ssdomg 8992	. . . . . . . . . 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 44557	. . . . . . . . . . . . 13 ⊢ ℚ ≼ ω
70	69, 69	pm3.2i 470	. . . . . . . . . . . 12 ⊢ (ℚ ≼ ω ∧ ℚ ≼ ω)
71		xpct 10007	. . . . . . . . . . . 12 ⊢ ((ℚ ≼ ω ∧ ℚ ≼ ω) → (ℚ × ℚ) ≼ ω)
72	70, 71	ax-mp 5	. . . . . . . . . . 11 ⊢ (ℚ × ℚ) ≼ ω
73		fimact 10526	. . . . . . . . . . 11 ⊢ (((ℚ × ℚ) ≼ ω ∧ Fun (,)) → ((,) “ (ℚ × ℚ)) ≼ ω)
74	72, 18, 73	mp2an 689	. . . . . . . . . 10 ⊢ ((,) “ (ℚ × ℚ)) ≼ ω
75	74	a1i 11	. . . . . . . . 9 ⊢ (𝑞 ⊆ ((,) “ (ℚ × ℚ)) → ((,) “ (ℚ × ℚ)) ≼ ω)
76		domtr 8999	. . . . . . . . 9 ⊢ ((𝑞 ≼ ((,) “ (ℚ × ℚ)) ∧ ((,) “ (ℚ × ℚ)) ≼ ω) → 𝑞 ≼ ω)
77	68, 75, 76	syl2anc 583	. . . . . . . 8 ⊢ (𝑞 ⊆ ((,) “ (ℚ × ℚ)) → 𝑞 ≼ ω)
78	77	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ⊆ ((,) “ (ℚ × ℚ))) → 𝑞 ≼ ω)
79	10, 66, 78	salunicl 45517	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ⊆ ((,) “ (ℚ × ℚ))) → ∪ 𝑞 ∈ 𝑇)
80	79	adantrr 714	. . . . 5 ⊢ ((𝜑 ∧ (𝑞 ⊆ ((,) “ (ℚ × ℚ)) ∧ 𝐺 = ∪ 𝑞)) → ∪ 𝑞 ∈ 𝑇)
81	4, 80	eqeltrd 2825	. . . 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 1928	. 2 ⊢ (𝜑 → (∃𝑞(𝑞 ⊆ ((,) “ (ℚ × ℚ)) ∧ 𝐺 = ∪ 𝑞) → 𝐺 ∈ 𝑇))
84	3, 83	mpd 15	1 ⊢ (𝜑 → 𝐺 ∈ 𝑇)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ∃wrex 3062 {crab 3424 Vcvv 3466 ⊆ wss 3940 𝒫 cpw 4594 ⟨cop 4626 ∪ cuni 4899 class class class wbr 5138 × cxp 5664 ^◡ccnv 5665 dom cdm 5666 ran crn 5667 “ cima 5669 Fun wfun 6527 ‘cfv 6533 (class class class)co 7401 ωcom 7848 1^st c1st 7966 2^nd c2nd 7967 ≼ cdom 8933 ℝcr 11105 ℝ^*cxr 11244 ℚcq 12929 (,)cioo 13321 ↾_t crest 17365 topGenctg 17382 SAlgcsalg 45509 SMblFncsmblfn 45896

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9632 ax-cc 10426 ax-ac2 10454 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-oadd 8465 df-omul 8466 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-acn 9933 df-ac 10107 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-n0 12470 df-z 12556 df-uz 12820 df-q 12930 df-rp 12972 df-ioo 13325 df-ico 13327 df-fl 13754 df-rest 17367 df-topgen 17388 df-bases 22771 df-salg 45510 df-smblfn 45897

This theorem is referenced by: smfpimbor1lem2 46000

W3C validator