stoweidlem35 - Mathbox for Glauco Siliprandi

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Theorem stoweidlem35 45656

Description: This lemma is used to prove the existence of a function p as in Lemma 1 of [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p_(t₀) = 0, and p > 0 on T - U. Here (𝑞‘𝑖) is used to represent p_(t_i) in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

**Hypotheses**
Ref	Expression
stoweidlem35.1	⊢ Ⅎ𝑡𝜑
stoweidlem35.2	⊢ Ⅎ𝑤𝜑
stoweidlem35.3	⊢ Ⅎℎ𝜑
stoweidlem35.4	⊢ 𝑄 = {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))}
stoweidlem35.5	⊢ 𝑊 = {𝑤 ∈ 𝐽 ∣ ∃ℎ ∈ 𝑄 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}}
stoweidlem35.6	⊢ 𝐺 = (𝑤 ∈ 𝑋 ↦ {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}})
stoweidlem35.7	⊢ (𝜑 → 𝐴 ∈ V)
stoweidlem35.8	⊢ (𝜑 → 𝑋 ∈ Fin)
stoweidlem35.9	⊢ (𝜑 → 𝑋 ⊆ 𝑊)
stoweidlem35.10	⊢ (𝜑 → (𝑇 ∖ 𝑈) ⊆ ∪ 𝑋)
stoweidlem35.11	⊢ (𝜑 → (𝑇 ∖ 𝑈) ≠ ∅)

**Assertion**
Ref	Expression
stoweidlem35	⊢ (𝜑 → ∃𝑚∃𝑞(𝑚 ∈ ℕ ∧ (𝑞:(1...𝑚)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)∃𝑖 ∈ (1...𝑚)0 < ((𝑞‘𝑖)‘𝑡))))

Distinct variable groups: ℎ,𝑖,𝑡,𝑤 𝑖,𝑚,𝑞,𝑡 𝑖,𝐺 𝑤,𝑄 𝑇,ℎ,𝑤 𝑈,𝑞 𝜑,𝑖,𝑚 𝐴,ℎ,𝑡 ℎ,𝑋,𝑖,𝑡,𝑤 𝑤,𝑚 𝑚,𝐺 𝑄,𝑞 𝑇,𝑞 𝑡,𝑍 𝑤,𝑈 Allowed substitution hints: 𝜑(𝑤,𝑡,ℎ,𝑞) 𝐴(𝑤,𝑖,𝑚,𝑞) 𝑄(𝑡,ℎ,𝑖,𝑚) 𝑇(𝑡,𝑖,𝑚) 𝑈(𝑡,ℎ,𝑖,𝑚) 𝐺(𝑤,𝑡,ℎ,𝑞) 𝐽(𝑤,𝑡,ℎ,𝑖,𝑚,𝑞) 𝑊(𝑤,𝑡,ℎ,𝑖,𝑚,𝑞) 𝑋(𝑚,𝑞) 𝑍(𝑤,ℎ,𝑖,𝑚,𝑞)

Proof of Theorem stoweidlem35 Dummy variables 𝑓 𝑔 𝑘 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		stoweidlem35.8	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ Fin)
2		stoweidlem35.6	. . . . . . . . . . 11 ⊢ 𝐺 = (𝑤 ∈ 𝑋 ↦ {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}})
3	2	rnmptfi 44778	. . . . . . . . . 10 ⊢ (𝑋 ∈ Fin → ran 𝐺 ∈ Fin)
4	1, 3	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐺 ∈ Fin)
5		fnchoice 44628	. . . . . . . . . . 11 ⊢ (ran 𝐺 ∈ Fin → ∃𝑔(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔‘𝑙) ∈ 𝑙)))
6	5	adantl 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ran 𝐺 ∈ Fin) → ∃𝑔(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔‘𝑙) ∈ 𝑙)))
7		simprl 769	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔‘𝑙) ∈ 𝑙))) → 𝑔 Fn ran 𝐺)
8		stoweidlem35.2	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑤𝜑
9		nfmpt1 5261	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑤(𝑤 ∈ 𝑋 ↦ {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}})
10	2, 9	nfcxfr 2890	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑤𝐺
11	10	nfrn 5958	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑤ran 𝐺
12	11	nfcri 2883	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑤 𝑘 ∈ ran 𝐺
13	8, 12	nfan 1895	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑤(𝜑 ∧ 𝑘 ∈ ran 𝐺)
14		stoweidlem35.9	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑋 ⊆ 𝑊)
15	14	sselda 3979	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑋) → 𝑤 ∈ 𝑊)
16		stoweidlem35.5	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 𝑊 = {𝑤 ∈ 𝐽 ∣ ∃ℎ ∈ 𝑄 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}}
17	15, 16	eleqtrdi 2836	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑋) → 𝑤 ∈ {𝑤 ∈ 𝐽 ∣ ∃ℎ ∈ 𝑄 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}})
18		rabid 3440	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑤 ∈ {𝑤 ∈ 𝐽 ∣ ∃ℎ ∈ 𝑄 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}} ↔ (𝑤 ∈ 𝐽 ∧ ∃ℎ ∈ 𝑄 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}))
19	17, 18	sylib 217	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑋) → (𝑤 ∈ 𝐽 ∧ ∃ℎ ∈ 𝑄 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}))
20	19	simprd 494	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑋) → ∃ℎ ∈ 𝑄 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)})
21		df-rex 3061	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (∃ℎ ∈ 𝑄 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)} ↔ ∃ℎ(ℎ ∈ 𝑄 ∧ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}))
22	20, 21	sylib 217	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑋) → ∃ℎ(ℎ ∈ 𝑄 ∧ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}))
23		rabid 3440	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ℎ ∈ {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}} ↔ (ℎ ∈ 𝑄 ∧ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}))
24	23	exbii 1843	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∃ℎ ℎ ∈ {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}} ↔ ∃ℎ(ℎ ∈ 𝑄 ∧ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}))
25	22, 24	sylibr 233	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑋) → ∃ℎ ℎ ∈ {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}})
26	25	adantr 479	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑋) ∧ 𝑘 = {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}}) → ∃ℎ ℎ ∈ {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}})
27		stoweidlem35.3	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎℎ𝜑
28		nfv 1910	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎℎ 𝑤 ∈ 𝑋
29	27, 28	nfan 1895	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎℎ(𝜑 ∧ 𝑤 ∈ 𝑋)
30		nfrab1 3439	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎℎ{ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}}
31	30	nfeq2 2910	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎℎ 𝑘 = {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}}
32	29, 31	nfan 1895	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎℎ((𝜑 ∧ 𝑤 ∈ 𝑋) ∧ 𝑘 = {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}})
33		eleq2 2815	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 = {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}} → (ℎ ∈ 𝑘 ↔ ℎ ∈ {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}}))
34	33	biimprd 247	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 = {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}} → (ℎ ∈ {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}} → ℎ ∈ 𝑘))
35	34	adantl 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑋) ∧ 𝑘 = {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}}) → (ℎ ∈ {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}} → ℎ ∈ 𝑘))
36	32, 35	eximd 2205	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑋) ∧ 𝑘 = {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}}) → (∃ℎ ℎ ∈ {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}} → ∃ℎ ℎ ∈ 𝑘))
37	26, 36	mpd 15	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑋) ∧ 𝑘 = {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}}) → ∃ℎ ℎ ∈ 𝑘)
38	37	adantllr 717	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑘 ∈ ran 𝐺) ∧ 𝑤 ∈ 𝑋) ∧ 𝑘 = {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}}) → ∃ℎ ℎ ∈ 𝑘)
39	2	elrnmpt 5962	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ ran 𝐺 → (𝑘 ∈ ran 𝐺 ↔ ∃𝑤 ∈ 𝑋 𝑘 = {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}}))
40	39	ibi 266	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ ran 𝐺 → ∃𝑤 ∈ 𝑋 𝑘 = {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}})
41	40	adantl 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ ran 𝐺) → ∃𝑤 ∈ 𝑋 𝑘 = {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}})
42	13, 38, 41	r19.29af 3256	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ ran 𝐺) → ∃ℎ ℎ ∈ 𝑘)
43		n0 4349	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ≠ ∅ ↔ ∃ℎ ℎ ∈ 𝑘)
44	42, 43	sylibr 233	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ ran 𝐺) → 𝑘 ≠ ∅)
45	44	adantlr 713	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔‘𝑙) ∈ 𝑙))) ∧ 𝑘 ∈ ran 𝐺) → 𝑘 ≠ ∅)
46		simplrr 776	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔‘𝑙) ∈ 𝑙))) ∧ 𝑘 ∈ ran 𝐺) → ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔‘𝑙) ∈ 𝑙))
47		neeq1 2993	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑙 = 𝑘 → (𝑙 ≠ ∅ ↔ 𝑘 ≠ ∅))
48		fveq2 6901	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑙 = 𝑘 → (𝑔‘𝑙) = (𝑔‘𝑘))
49	48	eleq1d 2811	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑙 = 𝑘 → ((𝑔‘𝑙) ∈ 𝑙 ↔ (𝑔‘𝑘) ∈ 𝑙))
50		eleq2 2815	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑙 = 𝑘 → ((𝑔‘𝑘) ∈ 𝑙 ↔ (𝑔‘𝑘) ∈ 𝑘))
51	49, 50	bitrd 278	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑙 = 𝑘 → ((𝑔‘𝑙) ∈ 𝑙 ↔ (𝑔‘𝑘) ∈ 𝑘))
52	47, 51	imbi12d 343	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑙 = 𝑘 → ((𝑙 ≠ ∅ → (𝑔‘𝑙) ∈ 𝑙) ↔ (𝑘 ≠ ∅ → (𝑔‘𝑘) ∈ 𝑘)))
53	52	rspccva 3607	. . . . . . . . . . . . . . . . . 18 ⊢ ((∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔‘𝑙) ∈ 𝑙) ∧ 𝑘 ∈ ran 𝐺) → (𝑘 ≠ ∅ → (𝑔‘𝑘) ∈ 𝑘))
54	46, 53	sylancom 586	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔‘𝑙) ∈ 𝑙))) ∧ 𝑘 ∈ ran 𝐺) → (𝑘 ≠ ∅ → (𝑔‘𝑘) ∈ 𝑘))
55	45, 54	mpd 15	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔‘𝑙) ∈ 𝑙))) ∧ 𝑘 ∈ ran 𝐺) → (𝑔‘𝑘) ∈ 𝑘)
56	55	ralrimiva 3136	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔‘𝑙) ∈ 𝑙))) → ∀𝑘 ∈ ran 𝐺(𝑔‘𝑘) ∈ 𝑘)
57		fveq2 6901	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑙 → (𝑔‘𝑘) = (𝑔‘𝑙))
58	57	eleq1d 2811	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑙 → ((𝑔‘𝑘) ∈ 𝑘 ↔ (𝑔‘𝑙) ∈ 𝑘))
59		eleq2 2815	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑙 → ((𝑔‘𝑙) ∈ 𝑘 ↔ (𝑔‘𝑙) ∈ 𝑙))
60	58, 59	bitrd 278	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑙 → ((𝑔‘𝑘) ∈ 𝑘 ↔ (𝑔‘𝑙) ∈ 𝑙))
61	60	cbvralvw 3225	. . . . . . . . . . . . . . 15 ⊢ (∀𝑘 ∈ ran 𝐺(𝑔‘𝑘) ∈ 𝑘 ↔ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙)
62	56, 61	sylib 217	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔‘𝑙) ∈ 𝑙))) → ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙)
63	7, 62	jca 510	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔‘𝑙) ∈ 𝑙))) → (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙))
64	63	ex 411	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔‘𝑙) ∈ 𝑙)) → (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙)))
65	64	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ran 𝐺 ∈ Fin) → ((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔‘𝑙) ∈ 𝑙)) → (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙)))
66	65	eximdv 1913	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ran 𝐺 ∈ Fin) → (∃𝑔(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔‘𝑙) ∈ 𝑙)) → ∃𝑔(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙)))
67	6, 66	mpd 15	. . . . . . . . 9 ⊢ ((𝜑 ∧ ran 𝐺 ∈ Fin) → ∃𝑔(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙))
68	4, 67	mpdan 685	. . . . . . . 8 ⊢ (𝜑 → ∃𝑔(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙))
69	68	ralrimivw 3140	. . . . . . 7 ⊢ (𝜑 → ∀𝑚 ∈ ℕ ∃𝑔(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙))
70		stoweidlem35.10	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑇 ∖ 𝑈) ⊆ ∪ 𝑋)
71		stoweidlem35.11	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑇 ∖ 𝑈) ≠ ∅)
72		ssn0 4405	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∖ 𝑈) ⊆ ∪ 𝑋 ∧ (𝑇 ∖ 𝑈) ≠ ∅) → ∪ 𝑋 ≠ ∅)
73	70, 71, 72	syl2anc 582	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ 𝑋 ≠ ∅)
74	73	neneqd 2935	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ ∪ 𝑋 = ∅)
75		unieq 4924	. . . . . . . . . . . 12 ⊢ (𝑋 = ∅ → ∪ 𝑋 = ∪ ∅)
76		uni0 4943	. . . . . . . . . . . 12 ⊢ ∪ ∅ = ∅
77	75, 76	eqtrdi 2782	. . . . . . . . . . 11 ⊢ (𝑋 = ∅ → ∪ 𝑋 = ∅)
78	74, 77	nsyl 140	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑋 = ∅)
79		dm0rn0 5931	. . . . . . . . . . 11 ⊢ (dom 𝐺 = ∅ ↔ ran 𝐺 = ∅)
80		stoweidlem35.4	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑄 = {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))}
81		stoweidlem35.7	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ∈ V)
82	80, 81	rabexd 5340	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑄 ∈ V)
83		nfrab1 3439	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎℎ{ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))}
84	80, 83	nfcxfr 2890	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎℎ𝑄
85	84	rabexgf 44623	. . . . . . . . . . . . . . . . 17 ⊢ (𝑄 ∈ V → {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}} ∈ V)
86	82, 85	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}} ∈ V)
87	86	adantr 479	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑋) → {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}} ∈ V)
88	8, 87, 2	fmptdf 7131	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺:𝑋⟶V)
89		dffn2 6730	. . . . . . . . . . . . . 14 ⊢ (𝐺 Fn 𝑋 ↔ 𝐺:𝑋⟶V)
90	88, 89	sylibr 233	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 Fn 𝑋)
91	90	fndmd 6665	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝐺 = 𝑋)
92	91	eqeq1d 2728	. . . . . . . . . . 11 ⊢ (𝜑 → (dom 𝐺 = ∅ ↔ 𝑋 = ∅))
93	79, 92	bitr3id 284	. . . . . . . . . 10 ⊢ (𝜑 → (ran 𝐺 = ∅ ↔ 𝑋 = ∅))
94	78, 93	mtbird 324	. . . . . . . . 9 ⊢ (𝜑 → ¬ ran 𝐺 = ∅)
95		fz1f1o 15714	. . . . . . . . . . 11 ⊢ (ran 𝐺 ∈ Fin → (ran 𝐺 = ∅ ∨ ((♯‘ran 𝐺) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘ran 𝐺))–1-1-onto→ran 𝐺)))
96	4, 95	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (ran 𝐺 = ∅ ∨ ((♯‘ran 𝐺) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘ran 𝐺))–1-1-onto→ran 𝐺)))
97	96	ord 862	. . . . . . . . 9 ⊢ (𝜑 → (¬ ran 𝐺 = ∅ → ((♯‘ran 𝐺) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘ran 𝐺))–1-1-onto→ran 𝐺)))
98	94, 97	mpd 15	. . . . . . . 8 ⊢ (𝜑 → ((♯‘ran 𝐺) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘ran 𝐺))–1-1-onto→ran 𝐺))
99		oveq2 7432	. . . . . . . . . . 11 ⊢ (𝑚 = (♯‘ran 𝐺) → (1...𝑚) = (1...(♯‘ran 𝐺)))
100	99	f1oeq2d 6839	. . . . . . . . . 10 ⊢ (𝑚 = (♯‘ran 𝐺) → (𝑓:(1...𝑚)–1-1-onto→ran 𝐺 ↔ 𝑓:(1...(♯‘ran 𝐺))–1-1-onto→ran 𝐺))
101	100	exbidv 1917	. . . . . . . . 9 ⊢ (𝑚 = (♯‘ran 𝐺) → (∃𝑓 𝑓:(1...𝑚)–1-1-onto→ran 𝐺 ↔ ∃𝑓 𝑓:(1...(♯‘ran 𝐺))–1-1-onto→ran 𝐺))
102	101	rspcev 3608	. . . . . . . 8 ⊢ (((♯‘ran 𝐺) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘ran 𝐺))–1-1-onto→ran 𝐺) → ∃𝑚 ∈ ℕ ∃𝑓 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)
103	98, 102	syl 17	. . . . . . 7 ⊢ (𝜑 → ∃𝑚 ∈ ℕ ∃𝑓 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)
104		r19.29 3104	. . . . . . 7 ⊢ ((∀𝑚 ∈ ℕ ∃𝑔(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙) ∧ ∃𝑚 ∈ ℕ ∃𝑓 𝑓:(1...𝑚)–1-1-onto→ran 𝐺) → ∃𝑚 ∈ ℕ (∃𝑔(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙) ∧ ∃𝑓 𝑓:(1...𝑚)–1-1-onto→ran 𝐺))
105	69, 103, 104	syl2anc 582	. . . . . 6 ⊢ (𝜑 → ∃𝑚 ∈ ℕ (∃𝑔(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙) ∧ ∃𝑓 𝑓:(1...𝑚)–1-1-onto→ran 𝐺))
106		exdistrv 1952	. . . . . . . . 9 ⊢ (∃𝑔∃𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺) ↔ (∃𝑔(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙) ∧ ∃𝑓 𝑓:(1...𝑚)–1-1-onto→ran 𝐺))
107	106	biimpri 227	. . . . . . . 8 ⊢ ((∃𝑔(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙) ∧ ∃𝑓 𝑓:(1...𝑚)–1-1-onto→ran 𝐺) → ∃𝑔∃𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺))
108	107	a1i 11	. . . . . . 7 ⊢ (𝜑 → ((∃𝑔(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙) ∧ ∃𝑓 𝑓:(1...𝑚)–1-1-onto→ran 𝐺) → ∃𝑔∃𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)))
109	108	reximdv 3160	. . . . . 6 ⊢ (𝜑 → (∃𝑚 ∈ ℕ (∃𝑔(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙) ∧ ∃𝑓 𝑓:(1...𝑚)–1-1-onto→ran 𝐺) → ∃𝑚 ∈ ℕ ∃𝑔∃𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)))
110	105, 109	mpd 15	. . . . 5 ⊢ (𝜑 → ∃𝑚 ∈ ℕ ∃𝑔∃𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺))
111		df-rex 3061	. . . . 5 ⊢ (∃𝑚 ∈ ℕ ∃𝑔∃𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺) ↔ ∃𝑚(𝑚 ∈ ℕ ∧ ∃𝑔∃𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)))
112	110, 111	sylib 217	. . . 4 ⊢ (𝜑 → ∃𝑚(𝑚 ∈ ℕ ∧ ∃𝑔∃𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)))
113		ax-5 1906	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → ∀𝑔 𝑚 ∈ ℕ)
114		19.29 1869	. . . . . . . . 9 ⊢ ((∀𝑔 𝑚 ∈ ℕ ∧ ∃𝑔∃𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)) → ∃𝑔(𝑚 ∈ ℕ ∧ ∃𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)))
115	113, 114	sylan 578	. . . . . . . 8 ⊢ ((𝑚 ∈ ℕ ∧ ∃𝑔∃𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)) → ∃𝑔(𝑚 ∈ ℕ ∧ ∃𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)))
116		ax-5 1906	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → ∀𝑓 𝑚 ∈ ℕ)
117		19.29 1869	. . . . . . . . . 10 ⊢ ((∀𝑓 𝑚 ∈ ℕ ∧ ∃𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)) → ∃𝑓(𝑚 ∈ ℕ ∧ ((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)))
118	116, 117	sylan 578	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ ∧ ∃𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)) → ∃𝑓(𝑚 ∈ ℕ ∧ ((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)))
119	118	eximi 1830	. . . . . . . 8 ⊢ (∃𝑔(𝑚 ∈ ℕ ∧ ∃𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)) → ∃𝑔∃𝑓(𝑚 ∈ ℕ ∧ ((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)))
120	115, 119	syl 17	. . . . . . 7 ⊢ ((𝑚 ∈ ℕ ∧ ∃𝑔∃𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)) → ∃𝑔∃𝑓(𝑚 ∈ ℕ ∧ ((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)))
121		df-3an 1086	. . . . . . . . 9 ⊢ ((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙 ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺) ↔ ((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺))
122	121	anbi2i 621	. . . . . . . 8 ⊢ ((𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙 ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)) ↔ (𝑚 ∈ ℕ ∧ ((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)))
123	122	2exbii 1844	. . . . . . 7 ⊢ (∃𝑔∃𝑓(𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙 ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)) ↔ ∃𝑔∃𝑓(𝑚 ∈ ℕ ∧ ((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)))
124	120, 123	sylibr 233	. . . . . 6 ⊢ ((𝑚 ∈ ℕ ∧ ∃𝑔∃𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)) → ∃𝑔∃𝑓(𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙 ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)))
125	124	a1i 11	. . . . 5 ⊢ (𝜑 → ((𝑚 ∈ ℕ ∧ ∃𝑔∃𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)) → ∃𝑔∃𝑓(𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙 ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺))))
126	125	eximdv 1913	. . . 4 ⊢ (𝜑 → (∃𝑚(𝑚 ∈ ℕ ∧ ∃𝑔∃𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)) → ∃𝑚∃𝑔∃𝑓(𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙 ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺))))
127	112, 126	mpd 15	. . 3 ⊢ (𝜑 → ∃𝑚∃𝑔∃𝑓(𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙 ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)))
128	82	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙 ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺))) → 𝑄 ∈ V)
129		simprl 769	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙 ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺))) → 𝑚 ∈ ℕ)
130		simprr1 1218	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙 ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺))) → 𝑔 Fn ran 𝐺)
131		elex 3482	. . . . . . . . 9 ⊢ (ran 𝐺 ∈ Fin → ran 𝐺 ∈ V)
132	4, 131	syl 17	. . . . . . . 8 ⊢ (𝜑 → ran 𝐺 ∈ V)
133	132	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙 ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺))) → ran 𝐺 ∈ V)
134		simprr2 1219	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙 ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺))) → ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙)
135	51	rspccva 3607	. . . . . . . 8 ⊢ ((∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙 ∧ 𝑘 ∈ ran 𝐺) → (𝑔‘𝑘) ∈ 𝑘)
136	134, 135	sylan 578	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙 ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺))) ∧ 𝑘 ∈ ran 𝐺) → (𝑔‘𝑘) ∈ 𝑘)
137		simprr3 1220	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙 ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺))) → 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)
138	70	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙 ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺))) → (𝑇 ∖ 𝑈) ⊆ ∪ 𝑋)
139		stoweidlem35.1	. . . . . . . 8 ⊢ Ⅎ𝑡𝜑
140		nfv 1910	. . . . . . . . 9 ⊢ Ⅎ𝑡 𝑚 ∈ ℕ
141		nfcv 2892	. . . . . . . . . . 11 ⊢ Ⅎ𝑡𝑔
142		nfcv 2892	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡𝑋
143		nfrab1 3439	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑡{𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}
144	143	nfeq2 2910	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑡 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}
145		nfv 1910	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑡(ℎ‘𝑍) = 0
146		nfra1 3272	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)
147	145, 146	nfan 1895	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑡((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))
148		nfcv 2892	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑡𝐴
149	147, 148	nfrabw 3457	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑡{ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))}
150	80, 149	nfcxfr 2890	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑡𝑄
151	144, 150	nfrabw 3457	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡{ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}}
152	142, 151	nfmpt 5260	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡(𝑤 ∈ 𝑋 ↦ {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}})
153	2, 152	nfcxfr 2890	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡𝐺
154	153	nfrn 5958	. . . . . . . . . . 11 ⊢ Ⅎ𝑡ran 𝐺
155	141, 154	nffn 6659	. . . . . . . . . 10 ⊢ Ⅎ𝑡 𝑔 Fn ran 𝐺
156		nfv 1910	. . . . . . . . . . 11 ⊢ Ⅎ𝑡(𝑔‘𝑙) ∈ 𝑙
157	154, 156	nfralw 3299	. . . . . . . . . 10 ⊢ Ⅎ𝑡∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙
158		nfcv 2892	. . . . . . . . . . 11 ⊢ Ⅎ𝑡𝑓
159		nfcv 2892	. . . . . . . . . . 11 ⊢ Ⅎ𝑡(1...𝑚)
160	158, 159, 154	nff1o 6841	. . . . . . . . . 10 ⊢ Ⅎ𝑡 𝑓:(1...𝑚)–1-1-onto→ran 𝐺
161	155, 157, 160	nf3an 1897	. . . . . . . . 9 ⊢ Ⅎ𝑡(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙 ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)
162	140, 161	nfan 1895	. . . . . . . 8 ⊢ Ⅎ𝑡(𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙 ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺))
163	139, 162	nfan 1895	. . . . . . 7 ⊢ Ⅎ𝑡(𝜑 ∧ (𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙 ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)))
164		nfv 1910	. . . . . . . . 9 ⊢ Ⅎ𝑤 𝑚 ∈ ℕ
165		nfcv 2892	. . . . . . . . . . 11 ⊢ Ⅎ𝑤𝑔
166	165, 11	nffn 6659	. . . . . . . . . 10 ⊢ Ⅎ𝑤 𝑔 Fn ran 𝐺
167		nfv 1910	. . . . . . . . . . 11 ⊢ Ⅎ𝑤(𝑔‘𝑙) ∈ 𝑙
168	11, 167	nfralw 3299	. . . . . . . . . 10 ⊢ Ⅎ𝑤∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙
169		nfcv 2892	. . . . . . . . . . 11 ⊢ Ⅎ𝑤𝑓
170		nfcv 2892	. . . . . . . . . . 11 ⊢ Ⅎ𝑤(1...𝑚)
171	169, 170, 11	nff1o 6841	. . . . . . . . . 10 ⊢ Ⅎ𝑤 𝑓:(1...𝑚)–1-1-onto→ran 𝐺
172	166, 168, 171	nf3an 1897	. . . . . . . . 9 ⊢ Ⅎ𝑤(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙 ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)
173	164, 172	nfan 1895	. . . . . . . 8 ⊢ Ⅎ𝑤(𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙 ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺))
174	8, 173	nfan 1895	. . . . . . 7 ⊢ Ⅎ𝑤(𝜑 ∧ (𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙 ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)))
175	2, 128, 129, 130, 133, 136, 137, 138, 163, 174, 84	stoweidlem27 45648	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙 ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺))) → ∃𝑞(𝑚 ∈ ℕ ∧ (𝑞:(1...𝑚)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)∃𝑖 ∈ (1...𝑚)0 < ((𝑞‘𝑖)‘𝑡))))
176	175	ex 411	. . . . 5 ⊢ (𝜑 → ((𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙 ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)) → ∃𝑞(𝑚 ∈ ℕ ∧ (𝑞:(1...𝑚)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)∃𝑖 ∈ (1...𝑚)0 < ((𝑞‘𝑖)‘𝑡)))))
177	176	2eximdv 1915	. . . 4 ⊢ (𝜑 → (∃𝑔∃𝑓(𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙 ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)) → ∃𝑔∃𝑓∃𝑞(𝑚 ∈ ℕ ∧ (𝑞:(1...𝑚)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)∃𝑖 ∈ (1...𝑚)0 < ((𝑞‘𝑖)‘𝑡)))))
178	177	eximdv 1913	. . 3 ⊢ (𝜑 → (∃𝑚∃𝑔∃𝑓(𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙 ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)) → ∃𝑚∃𝑔∃𝑓∃𝑞(𝑚 ∈ ℕ ∧ (𝑞:(1...𝑚)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)∃𝑖 ∈ (1...𝑚)0 < ((𝑞‘𝑖)‘𝑡)))))
179	127, 178	mpd 15	. 2 ⊢ (𝜑 → ∃𝑚∃𝑔∃𝑓∃𝑞(𝑚 ∈ ℕ ∧ (𝑞:(1...𝑚)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)∃𝑖 ∈ (1...𝑚)0 < ((𝑞‘𝑖)‘𝑡))))
180		id 22	. . . 4 ⊢ (∃𝑞(𝑚 ∈ ℕ ∧ (𝑞:(1...𝑚)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)∃𝑖 ∈ (1...𝑚)0 < ((𝑞‘𝑖)‘𝑡))) → ∃𝑞(𝑚 ∈ ℕ ∧ (𝑞:(1...𝑚)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)∃𝑖 ∈ (1...𝑚)0 < ((𝑞‘𝑖)‘𝑡))))
181	180	exlimivv 1928	. . 3 ⊢ (∃𝑔∃𝑓∃𝑞(𝑚 ∈ ℕ ∧ (𝑞:(1...𝑚)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)∃𝑖 ∈ (1...𝑚)0 < ((𝑞‘𝑖)‘𝑡))) → ∃𝑞(𝑚 ∈ ℕ ∧ (𝑞:(1...𝑚)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)∃𝑖 ∈ (1...𝑚)0 < ((𝑞‘𝑖)‘𝑡))))
182	181	eximi 1830	. 2 ⊢ (∃𝑚∃𝑔∃𝑓∃𝑞(𝑚 ∈ ℕ ∧ (𝑞:(1...𝑚)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)∃𝑖 ∈ (1...𝑚)0 < ((𝑞‘𝑖)‘𝑡))) → ∃𝑚∃𝑞(𝑚 ∈ ℕ ∧ (𝑞:(1...𝑚)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)∃𝑖 ∈ (1...𝑚)0 < ((𝑞‘𝑖)‘𝑡))))
183	179, 182	syl 17	1 ⊢ (𝜑 → ∃𝑚∃𝑞(𝑚 ∈ ℕ ∧ (𝑞:(1...𝑚)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)∃𝑖 ∈ (1...𝑚)0 < ((𝑞‘𝑖)‘𝑡))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∨ wo 845 ∧ w3a 1084 ∀wal 1532 = wceq 1534 ∃wex 1774 Ⅎwnf 1778 ∈ wcel 2099 ≠ wne 2930 ∀wral 3051 ∃wrex 3060 {crab 3419 Vcvv 3462 ∖ cdif 3944 ⊆ wss 3947 ∅c0 4325 ∪ cuni 4913 class class class wbr 5153 ↦ cmpt 5236 dom cdm 5682 ran crn 5683 Fn wfn 6549 ⟶wf 6550 –1-1-onto→wf1o 6553 ‘cfv 6554 (class class class)co 7424 Fincfn 8974 0cc0 11158 1c1 11159 < clt 11298 ≤ cle 11299 ℕcn 12264 ...cfz 13538 ♯chash 14347

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-n0 12525 df-z 12611 df-uz 12875 df-fz 13539 df-hash 14348

This theorem is referenced by: stoweidlem53 45674

W3C validator