Theorem stoweidlem35 41816

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 40920	. . . . . . . . . 10 ⊢ (𝑋 ∈ Fin → ran 𝐺 ∈ Fin)
4	1, 3	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐺 ∈ Fin)
5		fnchoice 40777	. . . . . . . . . . 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 767	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔‘𝑙) ∈ 𝑙))) → 𝑔 Fn ran 𝐺)
8		stoweidlem35.2	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑤𝜑
9		nfmpt1 5052	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑤(𝑤 ∈ 𝑋 ↦ {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}})
10	2, 9	nfcxfr 2945	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑤𝐺
11	10	nfrn 5698	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑤ran 𝐺
12	11	nfcri 2941	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑤 𝑘 ∈ ran 𝐺
13	8, 12	nfan 1879	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑤(𝜑 ∧ 𝑘 ∈ ran 𝐺)
14		stoweidlem35.9	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑋 ⊆ 𝑊)
15	14	sselda 3884	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑋) → 𝑤 ∈ 𝑊)
16		stoweidlem35.5	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 𝑊 = {𝑤 ∈ 𝐽 ∣ ∃ℎ ∈ 𝑄 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}}
17	15, 16	syl6eleq 2891	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑋) → 𝑤 ∈ {𝑤 ∈ 𝐽 ∣ ∃ℎ ∈ 𝑄 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}})
18		rabid 3334	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 219	. . . . . . . . . . . . . . . . . . . . . . . . . 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 3109	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (∃ℎ ∈ 𝑄 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)} ↔ ∃ℎ(ℎ ∈ 𝑄 ∧ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}))
22	20, 21	sylib 219	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑋) → ∃ℎ(ℎ ∈ 𝑄 ∧ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}))
23		rabid 3334	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ℎ ∈ {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}} ↔ (ℎ ∈ 𝑄 ∧ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}))
24	23	exbii 1827	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∃ℎ ℎ ∈ {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}} ↔ ∃ℎ(ℎ ∈ 𝑄 ∧ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}))
25	22, 24	sylibr 235	. . . . . . . . . . . . . . . . . . . . . . 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 1890	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎℎ 𝑤 ∈ 𝑋
29	27, 28	nfan 1879	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎℎ(𝜑 ∧ 𝑤 ∈ 𝑋)
30		nfrab1 3341	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎℎ{ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}}
31	30	nfeq2 2962	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎℎ 𝑘 = {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}}
32	29, 31	nfan 1879	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎℎ((𝜑 ∧ 𝑤 ∈ 𝑋) ∧ 𝑘 = {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}})
33		eleq2 2869	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 = {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}} → (ℎ ∈ 𝑘 ↔ ℎ ∈ {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}}))
34	33	biimprd 249	. . . . . . . . . . . . . . . . . . . . . . . 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 2179	. . . . . . . . . . . . . . . . . . . . . 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 715	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑘 ∈ ran 𝐺) ∧ 𝑤 ∈ 𝑋) ∧ 𝑘 = {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}}) → ∃ℎ ℎ ∈ 𝑘)
39	2	elrnmpt 5702	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ ran 𝐺 → (𝑘 ∈ ran 𝐺 ↔ ∃𝑤 ∈ 𝑋 𝑘 = {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}}))
40	39	ibi 268	. . . . . . . . . . . . . . . . . . . . 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 3289	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ ran 𝐺) → ∃ℎ ℎ ∈ 𝑘)
43		n0 4224	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ≠ ∅ ↔ ∃ℎ ℎ ∈ 𝑘)
44	42, 43	sylibr 235	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ ran 𝐺) → 𝑘 ≠ ∅)
45	44	adantlr 711	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔‘𝑙) ∈ 𝑙))) ∧ 𝑘 ∈ ran 𝐺) → 𝑘 ≠ ∅)
46		simplrr 774	. . . . . . . . . . . . . . . . . 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 3044	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑙 = 𝑘 → (𝑙 ≠ ∅ ↔ 𝑘 ≠ ∅))
48		fveq2 6530	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑙 = 𝑘 → (𝑔‘𝑙) = (𝑔‘𝑘))
49	48	eleq1d 2865	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑙 = 𝑘 → ((𝑔‘𝑙) ∈ 𝑙 ↔ (𝑔‘𝑘) ∈ 𝑙))
50		eleq2 2869	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑙 = 𝑘 → ((𝑔‘𝑘) ∈ 𝑙 ↔ (𝑔‘𝑘) ∈ 𝑘))
51	49, 50	bitrd 280	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑙 = 𝑘 → ((𝑔‘𝑙) ∈ 𝑙 ↔ (𝑔‘𝑘) ∈ 𝑘))
52	47, 51	imbi12d 346	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑙 = 𝑘 → ((𝑙 ≠ ∅ → (𝑔‘𝑙) ∈ 𝑙) ↔ (𝑘 ≠ ∅ → (𝑔‘𝑘) ∈ 𝑘)))
53	52	rspccva 3553	. . . . . . . . . . . . . . . . . 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 3147	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔‘𝑙) ∈ 𝑙))) → ∀𝑘 ∈ ran 𝐺(𝑔‘𝑘) ∈ 𝑘)
57		fveq2 6530	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑙 → (𝑔‘𝑘) = (𝑔‘𝑙))
58	57	eleq1d 2865	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑙 → ((𝑔‘𝑘) ∈ 𝑘 ↔ (𝑔‘𝑙) ∈ 𝑘))
59		eleq2 2869	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑙 → ((𝑔‘𝑙) ∈ 𝑘 ↔ (𝑔‘𝑙) ∈ 𝑙))
60	58, 59	bitrd 280	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑙 → ((𝑔‘𝑘) ∈ 𝑘 ↔ (𝑔‘𝑙) ∈ 𝑙))
61	60	cbvralv 3400	. . . . . . . . . . . . . . 15 ⊢ (∀𝑘 ∈ ran 𝐺(𝑔‘𝑘) ∈ 𝑘 ↔ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙)
62	56, 61	sylib 219	. . . . . . . . . . . . . 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 1893	. . . . . . . . . 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 683	. . . . . . . 8 ⊢ (𝜑 → ∃𝑔(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙))
69	68	ralrimivw 3148	. . . . . . 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 4268	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∖ 𝑈) ⊆ ∪ 𝑋 ∧ (𝑇 ∖ 𝑈) ≠ ∅) → ∪ 𝑋 ≠ ∅)
73	70, 71, 72	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ 𝑋 ≠ ∅)
74	73	neneqd 2987	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ ∪ 𝑋 = ∅)
75		unieq 4747	. . . . . . . . . . . 12 ⊢ (𝑋 = ∅ → ∪ 𝑋 = ∪ ∅)
76		uni0 4766	. . . . . . . . . . . 12 ⊢ ∪ ∅ = ∅
77	75, 76	syl6eq 2845	. . . . . . . . . . 11 ⊢ (𝑋 = ∅ → ∪ 𝑋 = ∅)
78	74, 77	nsyl 142	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑋 = ∅)
79		dm0rn0 5671	. . . . . . . . . . 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 5120	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑄 ∈ V)
83		nfrab1 3341	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎℎ{ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))}
84	80, 83	nfcxfr 2945	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎℎ𝑄
85	84	rabexgf 40772	. . . . . . . . . . . . . . . . 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 6735	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺:𝑋⟶V)
89		dffn2 6376	. . . . . . . . . . . . . 14 ⊢ (𝐺 Fn 𝑋 ↔ 𝐺:𝑋⟶V)
90	88, 89	sylibr 235	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 Fn 𝑋)
91		fndm 6317	. . . . . . . . . . . . 13 ⊢ (𝐺 Fn 𝑋 → dom 𝐺 = 𝑋)
92	90, 91	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝐺 = 𝑋)
93	92	eqeq1d 2795	. . . . . . . . . . 11 ⊢ (𝜑 → (dom 𝐺 = ∅ ↔ 𝑋 = ∅))
94	79, 93	syl5bbr 286	. . . . . . . . . 10 ⊢ (𝜑 → (ran 𝐺 = ∅ ↔ 𝑋 = ∅))
95	78, 94	mtbird 326	. . . . . . . . 9 ⊢ (𝜑 → ¬ ran 𝐺 = ∅)
96		fz1f1o 14888	. . . . . . . . . . 11 ⊢ (ran 𝐺 ∈ Fin → (ran 𝐺 = ∅ ∨ ((♯‘ran 𝐺) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘ran 𝐺))–1-1-onto→ran 𝐺)))
97	4, 96	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (ran 𝐺 = ∅ ∨ ((♯‘ran 𝐺) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘ran 𝐺))–1-1-onto→ran 𝐺)))
98	97	ord 859	. . . . . . . . 9 ⊢ (𝜑 → (¬ ran 𝐺 = ∅ → ((♯‘ran 𝐺) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘ran 𝐺))–1-1-onto→ran 𝐺)))
99	95, 98	mpd 15	. . . . . . . 8 ⊢ (𝜑 → ((♯‘ran 𝐺) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘ran 𝐺))–1-1-onto→ran 𝐺))
100		oveq2 7015	. . . . . . . . . . 11 ⊢ (𝑚 = (♯‘ran 𝐺) → (1...𝑚) = (1...(♯‘ran 𝐺)))
101	100	f1oeq2d 6471	. . . . . . . . . 10 ⊢ (𝑚 = (♯‘ran 𝐺) → (𝑓:(1...𝑚)–1-1-onto→ran 𝐺 ↔ 𝑓:(1...(♯‘ran 𝐺))–1-1-onto→ran 𝐺))
102	101	exbidv 1897	. . . . . . . . 9 ⊢ (𝑚 = (♯‘ran 𝐺) → (∃𝑓 𝑓:(1...𝑚)–1-1-onto→ran 𝐺 ↔ ∃𝑓 𝑓:(1...(♯‘ran 𝐺))–1-1-onto→ran 𝐺))
103	102	rspcev 3554	. . . . . . . 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)
104	99, 103	syl 17	. . . . . . 7 ⊢ (𝜑 → ∃𝑚 ∈ ℕ ∃𝑓 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)
105		r19.29 3215	. . . . . . 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))
106	69, 104, 105	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))
107		exdistrv 1931	. . . . . . . . 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))
108	107	biimpri 229	. . . . . . . 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))
109	108	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)))
110	109	reximdv 3233	. . . . . 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)))
111	106, 110	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))
112		df-rex 3109	. . . . 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)))
113	111, 112	sylib 219	. . . 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)))
114		ax-5 1886	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → ∀𝑔 𝑚 ∈ ℕ)
115		19.29 1853	. . . . . . . . 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)))
116	114, 115	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)))
117		ax-5 1886	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → ∀𝑓 𝑚 ∈ ℕ)
118		19.29 1853	. . . . . . . . . 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)))
119	117, 118	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)))
120	119	eximi 1814	. . . . . . . 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)))
121	116, 120	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)))
122		df-3an 1080	. . . . . . . . 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))
123	122	anbi2i 622	. . . . . . . 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)))
124	123	2exbii 1828	. . . . . . 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)))
125	121, 124	sylibr 235	. . . . . 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)))
126	125	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))))
127	126	eximdv 1893	. . . 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))))
128	113, 127	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)))
129	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)
130		simprl 767	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙 ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺))) → 𝑚 ∈ ℕ)
131		simprr1 1212	. . . . . . 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)
132		elex 3450	. . . . . . . . 9 ⊢ (ran 𝐺 ∈ Fin → ran 𝐺 ∈ V)
133	4, 132	syl 17	. . . . . . . 8 ⊢ (𝜑 → ran 𝐺 ∈ V)
134	133	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)
135		simprr2 1213	. . . . . . . 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)
136	51	rspccva 3553	. . . . . . . 8 ⊢ ((∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙 ∧ 𝑘 ∈ ran 𝐺) → (𝑔‘𝑘) ∈ 𝑘)
137	135, 136	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)
138		simprr3 1214	. . . . . . 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)
139	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)
140		stoweidlem35.1	. . . . . . . 8 ⊢ Ⅎ𝑡𝜑
141		nfv 1890	. . . . . . . . 9 ⊢ Ⅎ𝑡 𝑚 ∈ ℕ
142		nfcv 2947	. . . . . . . . . . 11 ⊢ Ⅎ𝑡𝑔
143		nfcv 2947	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡𝑋
144		nfrab1 3341	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑡{𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}
145	144	nfeq2 2962	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑡 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}
146		nfv 1890	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑡(ℎ‘𝑍) = 0
147		nfra1 3184	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)
148	146, 147	nfan 1879	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑡((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))
149		nfcv 2947	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑡𝐴
150	148, 149	nfrab 3342	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑡{ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))}
151	80, 150	nfcxfr 2945	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑡𝑄
152	145, 151	nfrab 3342	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡{ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}}
153	143, 152	nfmpt 5051	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡(𝑤 ∈ 𝑋 ↦ {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}})
154	2, 153	nfcxfr 2945	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡𝐺
155	154	nfrn 5698	. . . . . . . . . . 11 ⊢ Ⅎ𝑡ran 𝐺
156	142, 155	nffn 6314	. . . . . . . . . 10 ⊢ Ⅎ𝑡 𝑔 Fn ran 𝐺
157		nfv 1890	. . . . . . . . . . 11 ⊢ Ⅎ𝑡(𝑔‘𝑙) ∈ 𝑙
158	155, 157	nfral 3189	. . . . . . . . . 10 ⊢ Ⅎ𝑡∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙
159		nfcv 2947	. . . . . . . . . . 11 ⊢ Ⅎ𝑡𝑓
160		nfcv 2947	. . . . . . . . . . 11 ⊢ Ⅎ𝑡(1...𝑚)
161	159, 160, 155	nff1o 6473	. . . . . . . . . 10 ⊢ Ⅎ𝑡 𝑓:(1...𝑚)–1-1-onto→ran 𝐺
162	156, 158, 161	nf3an 1881	. . . . . . . . 9 ⊢ Ⅎ𝑡(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙 ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)
163	141, 162	nfan 1879	. . . . . . . 8 ⊢ Ⅎ𝑡(𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙 ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺))
164	140, 163	nfan 1879	. . . . . . 7 ⊢ Ⅎ𝑡(𝜑 ∧ (𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙 ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)))
165		nfv 1890	. . . . . . . . 9 ⊢ Ⅎ𝑤 𝑚 ∈ ℕ
166		nfcv 2947	. . . . . . . . . . 11 ⊢ Ⅎ𝑤𝑔
167	166, 11	nffn 6314	. . . . . . . . . 10 ⊢ Ⅎ𝑤 𝑔 Fn ran 𝐺
168		nfv 1890	. . . . . . . . . . 11 ⊢ Ⅎ𝑤(𝑔‘𝑙) ∈ 𝑙
169	11, 168	nfral 3189	. . . . . . . . . 10 ⊢ Ⅎ𝑤∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙
170		nfcv 2947	. . . . . . . . . . 11 ⊢ Ⅎ𝑤𝑓
171		nfcv 2947	. . . . . . . . . . 11 ⊢ Ⅎ𝑤(1...𝑚)
172	170, 171, 11	nff1o 6473	. . . . . . . . . 10 ⊢ Ⅎ𝑤 𝑓:(1...𝑚)–1-1-onto→ran 𝐺
173	167, 169, 172	nf3an 1881	. . . . . . . . 9 ⊢ Ⅎ𝑤(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙 ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)
174	165, 173	nfan 1879	. . . . . . . 8 ⊢ Ⅎ𝑤(𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙 ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺))
175	8, 174	nfan 1879	. . . . . . 7 ⊢ Ⅎ𝑤(𝜑 ∧ (𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔‘𝑙) ∈ 𝑙 ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)))
176	2, 129, 130, 131, 134, 137, 138, 139, 164, 175, 84	stoweidlem27 41808	. . . . . 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))))
177	176	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)))))
178	177	2eximdv 1895	. . . 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)))))
179	178	eximdv 1893	. . 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)))))
180	128, 179	mpd 15	. 2 ⊢ (𝜑 → ∃𝑚∃𝑔∃𝑓∃𝑞(𝑚 ∈ ℕ ∧ (𝑞:(1...𝑚)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)∃𝑖 ∈ (1...𝑚)0 < ((𝑞‘𝑖)‘𝑡))))
181		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))))
182	181	exlimivv 1908	. . 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))))
183	182	eximi 1814	. 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))))
184	180, 183	syl 17	1 ⊢ (𝜑 → ∃𝑚∃𝑞(𝑚 ∈ ℕ ∧ (𝑞:(1...𝑚)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)∃𝑖 ∈ (1...𝑚)0 < ((𝑞‘𝑖)‘𝑡))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∨ wo 842   ∧ w3a 1078  ∀wal 1518   = wceq 1520  ∃wex 1759  Ⅎwnf 1763   ∈ wcel 2079   ≠ wne 2982  ∀wral 3103  ∃wrex 3104  {crab 3107  Vcvv 3432   ∖ cdif 3851   ⊆ wss 3854  ∅c0 4206  ∪ cuni 4739   class class class wbr 4956   ↦ cmpt 5035  dom cdm 5435  ran crn 5436   Fn wfn 6212  ⟶wf 6213  –1-1-onto→wf1o 6216  ‘cfv 6217  (class class class)co 7007  Fincfn 8347  0cc0 10372  1c1 10373   < clt 10510   ≤ cle 10511  ℕcn 11475  ...cfz 12731  ♯chash 13528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-rep 5075  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310  ax-cnex 10428  ax-resscn 10429  ax-1cn 10430  ax-icn 10431  ax-addcl 10432  ax-addrcl 10433  ax-mulcl 10434  ax-mulrcl 10435  ax-mulcom 10436  ax-addass 10437  ax-mulass 10438  ax-distr 10439  ax-i2m1 10440  ax-1ne0 10441  ax-1rid 10442  ax-rnegex 10443  ax-rrecex 10444  ax-cnre 10445  ax-pre-lttri 10446  ax-pre-lttrn 10447  ax-pre-ltadd 10448  ax-pre-mulgt0 10449

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1079  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-nel 3089  df-ral 3108  df-rex 3109  df-reu 3110  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-pss 3871  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-tp 4471  df-op 4473  df-uni 4740  df-int 4777  df-iun 4821  df-br 4957  df-opab 5019  df-mpt 5036  df-tr 5058  df-id 5340  df-eprel 5345  df-po 5354  df-so 5355  df-fr 5394  df-we 5396  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-pred 6015  df-ord 6061  df-on 6062  df-lim 6063  df-suc 6064  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-fo 6223  df-f1o 6224  df-fv 6225  df-riota 6968  df-ov 7010  df-oprab 7011  df-mpo 7012  df-om 7428  df-1st 7536  df-2nd 7537  df-wrecs 7789  df-recs 7851  df-rdg 7889  df-1o 7944  df-oadd 7948  df-er 8130  df-en 8348  df-dom 8349  df-sdom 8350  df-fin 8351  df-card 9203  df-pnf 10512  df-mnf 10513  df-xr 10514  df-ltxr 10515  df-le 10516  df-sub 10708  df-neg 10709  df-nn 11476  df-n0 11735  df-z 11819  df-uz 12083  df-fz 12732  df-hash 13529

This theorem is referenced by:  stoweidlem53  41834