stoweidlem35 - Mathbox for Glauco Siliprandi

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Theorem stoweidlem35 43576

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 42707	. . . . . . . . . 10 ⊢ (𝑋 ∈ Fin → ran 𝐺 ∈ Fin)
4	1, 3	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐺 ∈ Fin)
5		fnchoice 42572	. . . . . . . . . . 11 ⊢ (ran 𝐺 ∈ Fin → ∃𝑔(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔‘𝑙) ∈ 𝑙)))
6	5	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ran 𝐺 ∈ Fin) → ∃𝑔(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔‘𝑙) ∈ 𝑙)))
7		simprl 768	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔‘𝑙) ∈ 𝑙))) → 𝑔 Fn ran 𝐺)
8		stoweidlem35.2	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑤𝜑
9		nfmpt1 5182	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑤(𝑤 ∈ 𝑋 ↦ {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}})
10	2, 9	nfcxfr 2905	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑤𝐺
11	10	nfrn 5861	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑤ran 𝐺
12	11	nfcri 2894	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑤 𝑘 ∈ ran 𝐺
13	8, 12	nfan 1902	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑤(𝜑 ∧ 𝑘 ∈ ran 𝐺)
14		stoweidlem35.9	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑋 ⊆ 𝑊)
15	14	sselda 3921	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2849	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑋) → 𝑤 ∈ {𝑤 ∈ 𝐽 ∣ ∃ℎ ∈ 𝑄 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}})
18		rabid 3310	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 496	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑋) → ∃ℎ ∈ 𝑄 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)})
21		df-rex 3070	. . . . . . . . . . . . . . . . . . . . . . . . 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 3310	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ℎ ∈ {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}} ↔ (ℎ ∈ 𝑄 ∧ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}))
24	23	exbii 1850	. . . . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . 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 1917	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎℎ 𝑤 ∈ 𝑋
29	27, 28	nfan 1902	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎℎ(𝜑 ∧ 𝑤 ∈ 𝑋)
30		nfrab1 3317	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎℎ{ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}}
31	30	nfeq2 2924	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎℎ 𝑘 = {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}}
32	29, 31	nfan 1902	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎℎ((𝜑 ∧ 𝑤 ∈ 𝑋) ∧ 𝑘 = {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}})
33		eleq2 2827	. . . . . . . . . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . . . . . . 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 2209	. . . . . . . . . . . . . . . . . . . . . 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 716	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑘 ∈ ran 𝐺) ∧ 𝑤 ∈ 𝑋) ∧ 𝑘 = {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}}) → ∃ℎ ℎ ∈ 𝑘)
39	2	elrnmpt 5865	. . . . . . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ ran 𝐺) → ∃𝑤 ∈ 𝑋 𝑘 = {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}})
42	13, 38, 41	r19.29af 3262	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ ran 𝐺) → ∃ℎ ℎ ∈ 𝑘)
43		n0 4280	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ≠ ∅ ↔ ∃ℎ ℎ ∈ 𝑘)
44	42, 43	sylibr 233	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ ran 𝐺) → 𝑘 ≠ ∅)
45	44	adantlr 712	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔‘𝑙) ∈ 𝑙))) ∧ 𝑘 ∈ ran 𝐺) → 𝑘 ≠ ∅)
46		simplrr 775	. . . . . . . . . . . . . . . . . 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 3006	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑙 = 𝑘 → (𝑙 ≠ ∅ ↔ 𝑘 ≠ ∅))
48		fveq2 6774	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑙 = 𝑘 → (𝑔‘𝑙) = (𝑔‘𝑘))
49	48	eleq1d 2823	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑙 = 𝑘 → ((𝑔‘𝑙) ∈ 𝑙 ↔ (𝑔‘𝑘) ∈ 𝑙))
50		eleq2 2827	. . . . . . . . . . . . . . . . . . . . 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 345	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑙 = 𝑘 → ((𝑙 ≠ ∅ → (𝑔‘𝑙) ∈ 𝑙) ↔ (𝑘 ≠ ∅ → (𝑔‘𝑘) ∈ 𝑘)))
53	52	rspccva 3560	. . . . . . . . . . . . . . . . . 18 ⊢ ((∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔‘𝑙) ∈ 𝑙) ∧ 𝑘 ∈ ran 𝐺) → (𝑘 ≠ ∅ → (𝑔‘𝑘) ∈ 𝑘))
54	46, 53	sylancom 588	. . . . . . . . . . . . . . . . 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 3103	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔‘𝑙) ∈ 𝑙))) → ∀𝑘 ∈ ran 𝐺(𝑔‘𝑘) ∈ 𝑘)
57		fveq2 6774	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑙 → (𝑔‘𝑘) = (𝑔‘𝑙))
58	57	eleq1d 2823	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑙 → ((𝑔‘𝑘) ∈ 𝑘 ↔ (𝑔‘𝑙) ∈ 𝑘))
59		eleq2 2827	. . . . . . . . . . . . . . . . 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 3383	. . . . . . . . . . . . . . 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 512	. . . . . . . . . . . . 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 413	. . . . . . . . . . . 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 481	. . . . . . . . . . 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 1920	. . . . . . . . . 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 684	. . . . . . . 8 ⊢ (𝜑 → ∃𝑔(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙))
69	68	ralrimivw 3104	. . . . . . 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 4334	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∖ 𝑈) ⊆ ∪ 𝑋 ∧ (𝑇 ∖ 𝑈) ≠ ∅) → ∪ 𝑋 ≠ ∅)
73	70, 71, 72	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ 𝑋 ≠ ∅)
74	73	neneqd 2948	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ ∪ 𝑋 = ∅)
75		unieq 4850	. . . . . . . . . . . 12 ⊢ (𝑋 = ∅ → ∪ 𝑋 = ∪ ∅)
76		uni0 4869	. . . . . . . . . . . 12 ⊢ ∪ ∅ = ∅
77	75, 76	eqtrdi 2794	. . . . . . . . . . 11 ⊢ (𝑋 = ∅ → ∪ 𝑋 = ∅)
78	74, 77	nsyl 140	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑋 = ∅)
79		dm0rn0 5834	. . . . . . . . . . 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 5257	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑄 ∈ V)
83		nfrab1 3317	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎℎ{ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))}
84	80, 83	nfcxfr 2905	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎℎ𝑄
85	84	rabexgf 42567	. . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑋) → {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}} ∈ V)
88	8, 87, 2	fmptdf 6991	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺:𝑋⟶V)
89		dffn2 6602	. . . . . . . . . . . . . 14 ⊢ (𝐺 Fn 𝑋 ↔ 𝐺:𝑋⟶V)
90	88, 89	sylibr 233	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 Fn 𝑋)
91	90	fndmd 6538	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝐺 = 𝑋)
92	91	eqeq1d 2740	. . . . . . . . . . 11 ⊢ (𝜑 → (dom 𝐺 = ∅ ↔ 𝑋 = ∅))
93	79, 92	bitr3id 285	. . . . . . . . . 10 ⊢ (𝜑 → (ran 𝐺 = ∅ ↔ 𝑋 = ∅))
94	78, 93	mtbird 325	. . . . . . . . 9 ⊢ (𝜑 → ¬ ran 𝐺 = ∅)
95		fz1f1o 15422	. . . . . . . . . . 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 861	. . . . . . . . 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 7283	. . . . . . . . . . 11 ⊢ (𝑚 = (♯‘ran 𝐺) → (1...𝑚) = (1...(♯‘ran 𝐺)))
100	99	f1oeq2d 6712	. . . . . . . . . 10 ⊢ (𝑚 = (♯‘ran 𝐺) → (𝑓:(1...𝑚)–1-1-onto→ran 𝐺 ↔ 𝑓:(1...(♯‘ran 𝐺))–1-1-onto→ran 𝐺))
101	100	exbidv 1924	. . . . . . . . 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 3561	. . . . . . . 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 3184	. . . . . . 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 584	. . . . . 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 1959	. . . . . . . . 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 3202	. . . . . 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 3070	. . . . 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 1913	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → ∀𝑔 𝑚 ∈ ℕ)
114		19.29 1876	. . . . . . . . 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 580	. . . . . . . 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 1913	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → ∀𝑓 𝑚 ∈ ℕ)
117		19.29 1876	. . . . . . . . . 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 580	. . . . . . . . 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 1837	. . . . . . . 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 1088	. . . . . . . . 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 623	. . . . . . . 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 1851	. . . . . . 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 1920	. . . 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 481	. . . . . . 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 768	. . . . . . 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 1220	. . . . . . 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 3450	. . . . . . . . 9 ⊢ (ran 𝐺 ∈ Fin → ran 𝐺 ∈ V)
132	4, 131	syl 17	. . . . . . . 8 ⊢ (𝜑 → ran 𝐺 ∈ V)
133	132	adantr 481	. . . . . . 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 1221	. . . . . . . 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 3560	. . . . . . . 8 ⊢ ((∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙 ∧ 𝑘 ∈ ran 𝐺) → (𝑔‘𝑘) ∈ 𝑘)
136	134, 135	sylan 580	. . . . . . 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 1222	. . . . . . 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 481	. . . . . . 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 1917	. . . . . . . . 9 ⊢ Ⅎ𝑡 𝑚 ∈ ℕ
141		nfcv 2907	. . . . . . . . . . 11 ⊢ Ⅎ𝑡𝑔
142		nfcv 2907	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡𝑋
143		nfrab1 3317	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑡{𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}
144	143	nfeq2 2924	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑡 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}
145		nfv 1917	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑡(ℎ‘𝑍) = 0
146		nfra1 3144	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)
147	145, 146	nfan 1902	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑡((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))
148		nfcv 2907	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑡𝐴
149	147, 148	nfrabw 3318	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑡{ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))}
150	80, 149	nfcxfr 2905	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑡𝑄
151	144, 150	nfrabw 3318	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡{ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}}
152	142, 151	nfmpt 5181	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡(𝑤 ∈ 𝑋 ↦ {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}})
153	2, 152	nfcxfr 2905	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡𝐺
154	153	nfrn 5861	. . . . . . . . . . 11 ⊢ Ⅎ𝑡ran 𝐺
155	141, 154	nffn 6532	. . . . . . . . . 10 ⊢ Ⅎ𝑡 𝑔 Fn ran 𝐺
156		nfv 1917	. . . . . . . . . . 11 ⊢ Ⅎ𝑡(𝑔‘𝑙) ∈ 𝑙
157	154, 156	nfralw 3151	. . . . . . . . . 10 ⊢ Ⅎ𝑡∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙
158		nfcv 2907	. . . . . . . . . . 11 ⊢ Ⅎ𝑡𝑓
159		nfcv 2907	. . . . . . . . . . 11 ⊢ Ⅎ𝑡(1...𝑚)
160	158, 159, 154	nff1o 6714	. . . . . . . . . 10 ⊢ Ⅎ𝑡 𝑓:(1...𝑚)–1-1-onto→ran 𝐺
161	155, 157, 160	nf3an 1904	. . . . . . . . 9 ⊢ Ⅎ𝑡(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙 ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)
162	140, 161	nfan 1902	. . . . . . . 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 1902	. . . . . . 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 1917	. . . . . . . . 9 ⊢ Ⅎ𝑤 𝑚 ∈ ℕ
165		nfcv 2907	. . . . . . . . . . 11 ⊢ Ⅎ𝑤𝑔
166	165, 11	nffn 6532	. . . . . . . . . 10 ⊢ Ⅎ𝑤 𝑔 Fn ran 𝐺
167		nfv 1917	. . . . . . . . . . 11 ⊢ Ⅎ𝑤(𝑔‘𝑙) ∈ 𝑙
168	11, 167	nfralw 3151	. . . . . . . . . 10 ⊢ Ⅎ𝑤∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙
169		nfcv 2907	. . . . . . . . . . 11 ⊢ Ⅎ𝑤𝑓
170		nfcv 2907	. . . . . . . . . . 11 ⊢ Ⅎ𝑤(1...𝑚)
171	169, 170, 11	nff1o 6714	. . . . . . . . . 10 ⊢ Ⅎ𝑤 𝑓:(1...𝑚)–1-1-onto→ran 𝐺
172	166, 168, 171	nf3an 1904	. . . . . . . . 9 ⊢ Ⅎ𝑤(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙 ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)
173	164, 172	nfan 1902	. . . . . . . 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 1902	. . . . . . 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 43568	. . . . . 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 413	. . . . 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 1922	. . . 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 1920	. . 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 1935	. . 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 1837	. 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 396 ∨ wo 844 ∧ w3a 1086 ∀wal 1537 = wceq 1539 ∃wex 1782 Ⅎwnf 1786 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3065 {crab 3068 Vcvv 3432 ∖ cdif 3884 ⊆ wss 3887 ∅c0 4256 ∪ cuni 4839 class class class wbr 5074 ↦ cmpt 5157 dom cdm 5589 ran crn 5590 Fn wfn 6428 ⟶wf 6429 –1-1-onto→wf1o 6432 ‘cfv 6433 (class class class)co 7275 Fincfn 8733 0cc0 10871 1c1 10872 < clt 11009 ≤ cle 11010 ℕcn 11973 ...cfz 13239 ♯chash 14044

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-hash 14045

This theorem is referenced by: stoweidlem53 43594

W3C validator