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Theorem stoweidlem35 40749
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(t_0) = 0, and p > 0 on T - U. Here (𝑞𝑖) is used to represent p(t_i) in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem35.1 𝑡𝜑
stoweidlem35.2 𝑤𝜑
stoweidlem35.3 𝜑
stoweidlem35.4 𝑄 = {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))}
stoweidlem35.5 𝑊 = {𝑤𝐽 ∣ ∃𝑄 𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}
stoweidlem35.6 𝐺 = (𝑤𝑋 ↦ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})
stoweidlem35.7 (𝜑𝐴 ∈ V)
stoweidlem35.8 (𝜑𝑋 ∈ Fin)
stoweidlem35.9 (𝜑𝑋𝑊)
stoweidlem35.10 (𝜑 → (𝑇𝑈) ⊆ 𝑋)
stoweidlem35.11 (𝜑 → (𝑇𝑈) ≠ ∅)
Assertion
Ref Expression
stoweidlem35 (𝜑 → ∃𝑚𝑞(𝑚 ∈ ℕ ∧ (𝑞:(1...𝑚)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇𝑈)∃𝑖 ∈ (1...𝑚)0 < ((𝑞𝑖)‘𝑡))))
Distinct variable groups:   ,𝑖,𝑡,𝑤   𝑖,𝑚,𝑞,𝑡   𝑖,𝐺   𝑤,𝑄   𝑇,,𝑤   𝑈,𝑞   𝜑,𝑖,𝑚   𝐴,,𝑡   ,𝑋,𝑖,𝑡,𝑤   𝑤,𝑚   𝑚,𝐺   𝑄,𝑞   𝑇,𝑞   𝑡,𝑍   𝑤,𝑈
Allowed substitution hints:   𝜑(𝑤,𝑡,,𝑞)   𝐴(𝑤,𝑖,𝑚,𝑞)   𝑄(𝑡,,𝑖,𝑚)   𝑇(𝑡,𝑖,𝑚)   𝑈(𝑡,,𝑖,𝑚)   𝐺(𝑤,𝑡,,𝑞)   𝐽(𝑤,𝑡,,𝑖,𝑚,𝑞)   𝑊(𝑤,𝑡,,𝑖,𝑚,𝑞)   𝑋(𝑚,𝑞)   𝑍(𝑤,,𝑖,𝑚,𝑞)

Proof of Theorem stoweidlem35
Dummy variables 𝑓 𝑔 𝑘 𝑙 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 stoweidlem35.8 . . . . . . . . . 10 (𝜑𝑋 ∈ Fin)
2 stoweidlem35.6 . . . . . . . . . . 11 𝐺 = (𝑤𝑋 ↦ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})
32rnmptfi 39858 . . . . . . . . . 10 (𝑋 ∈ Fin → ran 𝐺 ∈ Fin)
41, 3syl 17 . . . . . . . . 9 (𝜑 → ran 𝐺 ∈ Fin)
5 fnchoice 39700 . . . . . . . . . . 11 (ran 𝐺 ∈ Fin → ∃𝑔(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔𝑙) ∈ 𝑙)))
65adantl 469 . . . . . . . . . 10 ((𝜑 ∧ ran 𝐺 ∈ Fin) → ∃𝑔(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔𝑙) ∈ 𝑙)))
7 simprl 778 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔𝑙) ∈ 𝑙))) → 𝑔 Fn ran 𝐺)
8 stoweidlem35.2 . . . . . . . . . . . . . . . . . . . . 21 𝑤𝜑
9 nfmpt1 4952 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑤(𝑤𝑋 ↦ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})
102, 9nfcxfr 2957 . . . . . . . . . . . . . . . . . . . . . . 23 𝑤𝐺
1110nfrn 5583 . . . . . . . . . . . . . . . . . . . . . 22 𝑤ran 𝐺
1211nfcri 2953 . . . . . . . . . . . . . . . . . . . . 21 𝑤 𝑘 ∈ ran 𝐺
138, 12nfan 1990 . . . . . . . . . . . . . . . . . . . 20 𝑤(𝜑𝑘 ∈ ran 𝐺)
14 stoweidlem35.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑𝑋𝑊)
1514sselda 3809 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑤𝑋) → 𝑤𝑊)
16 stoweidlem35.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑊 = {𝑤𝐽 ∣ ∃𝑄 𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}
1715, 16syl6eleq 2906 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑤𝑋) → 𝑤 ∈ {𝑤𝐽 ∣ ∃𝑄 𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})
18 rabid 3315 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑤 ∈ {𝑤𝐽 ∣ ∃𝑄 𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} ↔ (𝑤𝐽 ∧ ∃𝑄 𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}))
1917, 18sylib 209 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑤𝑋) → (𝑤𝐽 ∧ ∃𝑄 𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}))
2019simprd 485 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑤𝑋) → ∃𝑄 𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)})
21 df-rex 3113 . . . . . . . . . . . . . . . . . . . . . . . . 25 (∃𝑄 𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)} ↔ ∃(𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}))
2220, 21sylib 209 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑤𝑋) → ∃(𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}))
23 rabid 3315 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( ∈ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} ↔ (𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}))
2423exbii 1933 . . . . . . . . . . . . . . . . . . . . . . . 24 (∃ ∈ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} ↔ ∃(𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}))
2522, 24sylibr 225 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑤𝑋) → ∃ ∈ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})
2625adantr 468 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑤𝑋) ∧ 𝑘 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}) → ∃ ∈ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})
27 stoweidlem35.3 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝜑
28 nfv 2005 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑤𝑋
2927, 28nfan 1990 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝑤𝑋)
30 nfrab1 3322 . . . . . . . . . . . . . . . . . . . . . . . . 25 {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}
3130nfeq2 2975 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}
3229, 31nfan 1990 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑤𝑋) ∧ 𝑘 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})
33 eleq2 2885 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} → (𝑘 ∈ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}))
3433biimprd 239 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} → ( ∈ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} → 𝑘))
3534adantl 469 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑤𝑋) ∧ 𝑘 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}) → ( ∈ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} → 𝑘))
3632, 35eximd 2253 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑤𝑋) ∧ 𝑘 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}) → (∃ ∈ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} → ∃ 𝑘))
3726, 36mpd 15 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑤𝑋) ∧ 𝑘 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}) → ∃ 𝑘)
3837adantllr 701 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑘 ∈ ran 𝐺) ∧ 𝑤𝑋) ∧ 𝑘 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}) → ∃ 𝑘)
392elrnmpt 5587 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ ran 𝐺 → (𝑘 ∈ ran 𝐺 ↔ ∃𝑤𝑋 𝑘 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}))
4039ibi 258 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ ran 𝐺 → ∃𝑤𝑋 𝑘 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})
4140adantl 469 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘 ∈ ran 𝐺) → ∃𝑤𝑋 𝑘 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})
4213, 38, 41r19.29af 3275 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ ran 𝐺) → ∃ 𝑘)
43 n0 4143 . . . . . . . . . . . . . . . . . . 19 (𝑘 ≠ ∅ ↔ ∃ 𝑘)
4442, 43sylibr 225 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ ran 𝐺) → 𝑘 ≠ ∅)
4544adantlr 697 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔𝑙) ∈ 𝑙))) ∧ 𝑘 ∈ ran 𝐺) → 𝑘 ≠ ∅)
46 simplrr 787 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔𝑙) ∈ 𝑙))) ∧ 𝑘 ∈ ran 𝐺) → ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔𝑙) ∈ 𝑙))
47 neeq1 3051 . . . . . . . . . . . . . . . . . . . 20 (𝑙 = 𝑘 → (𝑙 ≠ ∅ ↔ 𝑘 ≠ ∅))
48 fveq2 6418 . . . . . . . . . . . . . . . . . . . . . 22 (𝑙 = 𝑘 → (𝑔𝑙) = (𝑔𝑘))
4948eleq1d 2881 . . . . . . . . . . . . . . . . . . . . 21 (𝑙 = 𝑘 → ((𝑔𝑙) ∈ 𝑙 ↔ (𝑔𝑘) ∈ 𝑙))
50 eleq2 2885 . . . . . . . . . . . . . . . . . . . . 21 (𝑙 = 𝑘 → ((𝑔𝑘) ∈ 𝑙 ↔ (𝑔𝑘) ∈ 𝑘))
5149, 50bitrd 270 . . . . . . . . . . . . . . . . . . . 20 (𝑙 = 𝑘 → ((𝑔𝑙) ∈ 𝑙 ↔ (𝑔𝑘) ∈ 𝑘))
5247, 51imbi12d 335 . . . . . . . . . . . . . . . . . . 19 (𝑙 = 𝑘 → ((𝑙 ≠ ∅ → (𝑔𝑙) ∈ 𝑙) ↔ (𝑘 ≠ ∅ → (𝑔𝑘) ∈ 𝑘)))
5352rspccva 3512 . . . . . . . . . . . . . . . . . 18 ((∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔𝑙) ∈ 𝑙) ∧ 𝑘 ∈ ran 𝐺) → (𝑘 ≠ ∅ → (𝑔𝑘) ∈ 𝑘))
5446, 53sylancom 578 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔𝑙) ∈ 𝑙))) ∧ 𝑘 ∈ ran 𝐺) → (𝑘 ≠ ∅ → (𝑔𝑘) ∈ 𝑘))
5545, 54mpd 15 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔𝑙) ∈ 𝑙))) ∧ 𝑘 ∈ ran 𝐺) → (𝑔𝑘) ∈ 𝑘)
5655ralrimiva 3165 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔𝑙) ∈ 𝑙))) → ∀𝑘 ∈ ran 𝐺(𝑔𝑘) ∈ 𝑘)
57 fveq2 6418 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑙 → (𝑔𝑘) = (𝑔𝑙))
5857eleq1d 2881 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑙 → ((𝑔𝑘) ∈ 𝑘 ↔ (𝑔𝑙) ∈ 𝑘))
59 eleq2 2885 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑙 → ((𝑔𝑙) ∈ 𝑘 ↔ (𝑔𝑙) ∈ 𝑙))
6058, 59bitrd 270 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑙 → ((𝑔𝑘) ∈ 𝑘 ↔ (𝑔𝑙) ∈ 𝑙))
6160cbvralv 3371 . . . . . . . . . . . . . . 15 (∀𝑘 ∈ ran 𝐺(𝑔𝑘) ∈ 𝑘 ↔ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙)
6256, 61sylib 209 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔𝑙) ∈ 𝑙))) → ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙)
637, 62jca 503 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔𝑙) ∈ 𝑙))) → (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙))
6463ex 399 . . . . . . . . . . . 12 (𝜑 → ((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔𝑙) ∈ 𝑙)) → (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙)))
6564adantr 468 . . . . . . . . . . 11 ((𝜑 ∧ ran 𝐺 ∈ Fin) → ((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔𝑙) ∈ 𝑙)) → (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙)))
6665eximdv 2008 . . . . . . . . . 10 ((𝜑 ∧ ran 𝐺 ∈ Fin) → (∃𝑔(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔𝑙) ∈ 𝑙)) → ∃𝑔(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙)))
676, 66mpd 15 . . . . . . . . 9 ((𝜑 ∧ ran 𝐺 ∈ Fin) → ∃𝑔(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙))
684, 67mpdan 670 . . . . . . . 8 (𝜑 → ∃𝑔(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙))
6968ralrimivw 3166 . . . . . . 7 (𝜑 → ∀𝑚 ∈ ℕ ∃𝑔(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙))
70 stoweidlem35.10 . . . . . . . . . . . . 13 (𝜑 → (𝑇𝑈) ⊆ 𝑋)
71 stoweidlem35.11 . . . . . . . . . . . . 13 (𝜑 → (𝑇𝑈) ≠ ∅)
72 ssn0 4185 . . . . . . . . . . . . 13 (((𝑇𝑈) ⊆ 𝑋 ∧ (𝑇𝑈) ≠ ∅) → 𝑋 ≠ ∅)
7370, 71, 72syl2anc 575 . . . . . . . . . . . 12 (𝜑 𝑋 ≠ ∅)
7473neneqd 2994 . . . . . . . . . . 11 (𝜑 → ¬ 𝑋 = ∅)
75 unieq 4649 . . . . . . . . . . . 12 (𝑋 = ∅ → 𝑋 = ∅)
76 uni0 4670 . . . . . . . . . . . 12 ∅ = ∅
7775, 76syl6eq 2867 . . . . . . . . . . 11 (𝑋 = ∅ → 𝑋 = ∅)
7874, 77nsyl 137 . . . . . . . . . 10 (𝜑 → ¬ 𝑋 = ∅)
79 dm0rn0 5557 . . . . . . . . . . 11 (dom 𝐺 = ∅ ↔ ran 𝐺 = ∅)
80 stoweidlem35.4 . . . . . . . . . . . . . . . . . 18 𝑄 = {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))}
81 stoweidlem35.7 . . . . . . . . . . . . . . . . . 18 (𝜑𝐴 ∈ V)
8280, 81rabexd 5021 . . . . . . . . . . . . . . . . 17 (𝜑𝑄 ∈ V)
83 nfrab1 3322 . . . . . . . . . . . . . . . . . . 19 {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))}
8480, 83nfcxfr 2957 . . . . . . . . . . . . . . . . . 18 𝑄
8584rabexgf 39695 . . . . . . . . . . . . . . . . 17 (𝑄 ∈ V → {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} ∈ V)
8682, 85syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} ∈ V)
8786adantr 468 . . . . . . . . . . . . . . 15 ((𝜑𝑤𝑋) → {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} ∈ V)
888, 87, 2fmptdf 6619 . . . . . . . . . . . . . 14 (𝜑𝐺:𝑋⟶V)
89 dffn2 6268 . . . . . . . . . . . . . 14 (𝐺 Fn 𝑋𝐺:𝑋⟶V)
9088, 89sylibr 225 . . . . . . . . . . . . 13 (𝜑𝐺 Fn 𝑋)
91 fndm 6211 . . . . . . . . . . . . 13 (𝐺 Fn 𝑋 → dom 𝐺 = 𝑋)
9290, 91syl 17 . . . . . . . . . . . 12 (𝜑 → dom 𝐺 = 𝑋)
9392eqeq1d 2819 . . . . . . . . . . 11 (𝜑 → (dom 𝐺 = ∅ ↔ 𝑋 = ∅))
9479, 93syl5bbr 276 . . . . . . . . . 10 (𝜑 → (ran 𝐺 = ∅ ↔ 𝑋 = ∅))
9578, 94mtbird 316 . . . . . . . . 9 (𝜑 → ¬ ran 𝐺 = ∅)
96 fz1f1o 14684 . . . . . . . . . . 11 (ran 𝐺 ∈ Fin → (ran 𝐺 = ∅ ∨ ((♯‘ran 𝐺) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘ran 𝐺))–1-1-onto→ran 𝐺)))
974, 96syl 17 . . . . . . . . . 10 (𝜑 → (ran 𝐺 = ∅ ∨ ((♯‘ran 𝐺) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘ran 𝐺))–1-1-onto→ran 𝐺)))
9897ord 882 . . . . . . . . 9 (𝜑 → (¬ ran 𝐺 = ∅ → ((♯‘ran 𝐺) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘ran 𝐺))–1-1-onto→ran 𝐺)))
9995, 98mpd 15 . . . . . . . 8 (𝜑 → ((♯‘ran 𝐺) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘ran 𝐺))–1-1-onto→ran 𝐺))
100 oveq2 6892 . . . . . . . . . . 11 (𝑚 = (♯‘ran 𝐺) → (1...𝑚) = (1...(♯‘ran 𝐺)))
101 f1oeq2 6354 . . . . . . . . . . 11 ((1...𝑚) = (1...(♯‘ran 𝐺)) → (𝑓:(1...𝑚)–1-1-onto→ran 𝐺𝑓:(1...(♯‘ran 𝐺))–1-1-onto→ran 𝐺))
102100, 101syl 17 . . . . . . . . . 10 (𝑚 = (♯‘ran 𝐺) → (𝑓:(1...𝑚)–1-1-onto→ran 𝐺𝑓:(1...(♯‘ran 𝐺))–1-1-onto→ran 𝐺))
103102exbidv 2012 . . . . . . . . 9 (𝑚 = (♯‘ran 𝐺) → (∃𝑓 𝑓:(1...𝑚)–1-1-onto→ran 𝐺 ↔ ∃𝑓 𝑓:(1...(♯‘ran 𝐺))–1-1-onto→ran 𝐺))
104103rspcev 3513 . . . . . . . 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)
10599, 104syl 17 . . . . . . 7 (𝜑 → ∃𝑚 ∈ ℕ ∃𝑓 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)
106 r19.29 3271 . . . . . . 7 ((∀𝑚 ∈ ℕ ∃𝑔(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙) ∧ ∃𝑚 ∈ ℕ ∃𝑓 𝑓:(1...𝑚)–1-1-onto→ran 𝐺) → ∃𝑚 ∈ ℕ (∃𝑔(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙) ∧ ∃𝑓 𝑓:(1...𝑚)–1-1-onto→ran 𝐺))
10769, 105, 106syl2anc 575 . . . . . 6 (𝜑 → ∃𝑚 ∈ ℕ (∃𝑔(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙) ∧ ∃𝑓 𝑓:(1...𝑚)–1-1-onto→ran 𝐺))
108 eeanv 2358 . . . . . . . . 9 (∃𝑔𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺) ↔ (∃𝑔(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙) ∧ ∃𝑓 𝑓:(1...𝑚)–1-1-onto→ran 𝐺))
109108biimpri 219 . . . . . . . 8 ((∃𝑔(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙) ∧ ∃𝑓 𝑓:(1...𝑚)–1-1-onto→ran 𝐺) → ∃𝑔𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺))
110109a1i 11 . . . . . . 7 (𝜑 → ((∃𝑔(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙) ∧ ∃𝑓 𝑓:(1...𝑚)–1-1-onto→ran 𝐺) → ∃𝑔𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)))
111110reximdv 3214 . . . . . 6 (𝜑 → (∃𝑚 ∈ ℕ (∃𝑔(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙) ∧ ∃𝑓 𝑓:(1...𝑚)–1-1-onto→ran 𝐺) → ∃𝑚 ∈ ℕ ∃𝑔𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)))
112107, 111mpd 15 . . . . 5 (𝜑 → ∃𝑚 ∈ ℕ ∃𝑔𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺))
113 df-rex 3113 . . . . 5 (∃𝑚 ∈ ℕ ∃𝑔𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺) ↔ ∃𝑚(𝑚 ∈ ℕ ∧ ∃𝑔𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)))
114112, 113sylib 209 . . . 4 (𝜑 → ∃𝑚(𝑚 ∈ ℕ ∧ ∃𝑔𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)))
115 ax-5 2001 . . . . . . . . 9 (𝑚 ∈ ℕ → ∀𝑔 𝑚 ∈ ℕ)
116 19.29 1963 . . . . . . . . 9 ((∀𝑔 𝑚 ∈ ℕ ∧ ∃𝑔𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)) → ∃𝑔(𝑚 ∈ ℕ ∧ ∃𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)))
117115, 116sylan 571 . . . . . . . 8 ((𝑚 ∈ ℕ ∧ ∃𝑔𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)) → ∃𝑔(𝑚 ∈ ℕ ∧ ∃𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)))
118 ax-5 2001 . . . . . . . . . 10 (𝑚 ∈ ℕ → ∀𝑓 𝑚 ∈ ℕ)
119 19.29 1963 . . . . . . . . . 10 ((∀𝑓 𝑚 ∈ ℕ ∧ ∃𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)) → ∃𝑓(𝑚 ∈ ℕ ∧ ((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)))
120118, 119sylan 571 . . . . . . . . 9 ((𝑚 ∈ ℕ ∧ ∃𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)) → ∃𝑓(𝑚 ∈ ℕ ∧ ((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)))
121120eximi 1919 . . . . . . . 8 (∃𝑔(𝑚 ∈ ℕ ∧ ∃𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)) → ∃𝑔𝑓(𝑚 ∈ ℕ ∧ ((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)))
122117, 121syl 17 . . . . . . 7 ((𝑚 ∈ ℕ ∧ ∃𝑔𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)) → ∃𝑔𝑓(𝑚 ∈ ℕ ∧ ((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)))
123 df-3an 1102 . . . . . . . . 9 ((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙𝑓:(1...𝑚)–1-1-onto→ran 𝐺) ↔ ((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺))
124123anbi2i 611 . . . . . . . 8 ((𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙𝑓:(1...𝑚)–1-1-onto→ran 𝐺)) ↔ (𝑚 ∈ ℕ ∧ ((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)))
1251242exbii 1934 . . . . . . 7 (∃𝑔𝑓(𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙𝑓:(1...𝑚)–1-1-onto→ran 𝐺)) ↔ ∃𝑔𝑓(𝑚 ∈ ℕ ∧ ((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)))
126122, 125sylibr 225 . . . . . 6 ((𝑚 ∈ ℕ ∧ ∃𝑔𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)) → ∃𝑔𝑓(𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙𝑓:(1...𝑚)–1-1-onto→ran 𝐺)))
127126a1i 11 . . . . 5 (𝜑 → ((𝑚 ∈ ℕ ∧ ∃𝑔𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)) → ∃𝑔𝑓(𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙𝑓:(1...𝑚)–1-1-onto→ran 𝐺))))
128127eximdv 2008 . . . 4 (𝜑 → (∃𝑚(𝑚 ∈ ℕ ∧ ∃𝑔𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)) → ∃𝑚𝑔𝑓(𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙𝑓:(1...𝑚)–1-1-onto→ran 𝐺))))
129114, 128mpd 15 . . 3 (𝜑 → ∃𝑚𝑔𝑓(𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙𝑓:(1...𝑚)–1-1-onto→ran 𝐺)))
13082adantr 468 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙𝑓:(1...𝑚)–1-1-onto→ran 𝐺))) → 𝑄 ∈ V)
131 simprl 778 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙𝑓:(1...𝑚)–1-1-onto→ran 𝐺))) → 𝑚 ∈ ℕ)
132 simprr1 1280 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙𝑓:(1...𝑚)–1-1-onto→ran 𝐺))) → 𝑔 Fn ran 𝐺)
133 elex 3417 . . . . . . . . 9 (ran 𝐺 ∈ Fin → ran 𝐺 ∈ V)
1344, 133syl 17 . . . . . . . 8 (𝜑 → ran 𝐺 ∈ V)
135134adantr 468 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙𝑓:(1...𝑚)–1-1-onto→ran 𝐺))) → ran 𝐺 ∈ V)
136 simprr2 1282 . . . . . . . 8 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙𝑓:(1...𝑚)–1-1-onto→ran 𝐺))) → ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙)
13751rspccva 3512 . . . . . . . 8 ((∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙𝑘 ∈ ran 𝐺) → (𝑔𝑘) ∈ 𝑘)
138136, 137sylan 571 . . . . . . 7 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙𝑓:(1...𝑚)–1-1-onto→ran 𝐺))) ∧ 𝑘 ∈ ran 𝐺) → (𝑔𝑘) ∈ 𝑘)
139 simprr3 1284 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙𝑓:(1...𝑚)–1-1-onto→ran 𝐺))) → 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)
14070adantr 468 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙𝑓:(1...𝑚)–1-1-onto→ran 𝐺))) → (𝑇𝑈) ⊆ 𝑋)
141 stoweidlem35.1 . . . . . . . 8 𝑡𝜑
142 nfv 2005 . . . . . . . . 9 𝑡 𝑚 ∈ ℕ
143 nfcv 2959 . . . . . . . . . . 11 𝑡𝑔
144 nfcv 2959 . . . . . . . . . . . . . 14 𝑡𝑋
145 nfrab1 3322 . . . . . . . . . . . . . . . 16 𝑡{𝑡𝑇 ∣ 0 < (𝑡)}
146145nfeq2 2975 . . . . . . . . . . . . . . 15 𝑡 𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}
147 nfv 2005 . . . . . . . . . . . . . . . . . 18 𝑡(𝑍) = 0
148 nfra1 3140 . . . . . . . . . . . . . . . . . 18 𝑡𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)
149147, 148nfan 1990 . . . . . . . . . . . . . . . . 17 𝑡((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))
150 nfcv 2959 . . . . . . . . . . . . . . . . 17 𝑡𝐴
151149, 150nfrab 3323 . . . . . . . . . . . . . . . 16 𝑡{𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))}
15280, 151nfcxfr 2957 . . . . . . . . . . . . . . 15 𝑡𝑄
153146, 152nfrab 3323 . . . . . . . . . . . . . 14 𝑡{𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}
154144, 153nfmpt 4951 . . . . . . . . . . . . 13 𝑡(𝑤𝑋 ↦ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})
1552, 154nfcxfr 2957 . . . . . . . . . . . 12 𝑡𝐺
156155nfrn 5583 . . . . . . . . . . 11 𝑡ran 𝐺
157143, 156nffn 6208 . . . . . . . . . 10 𝑡 𝑔 Fn ran 𝐺
158 nfv 2005 . . . . . . . . . . 11 𝑡(𝑔𝑙) ∈ 𝑙
159156, 158nfral 3144 . . . . . . . . . 10 𝑡𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙
160 nfcv 2959 . . . . . . . . . . 11 𝑡𝑓
161 nfcv 2959 . . . . . . . . . . 11 𝑡(1...𝑚)
162160, 161, 156nff1o 6361 . . . . . . . . . 10 𝑡 𝑓:(1...𝑚)–1-1-onto→ran 𝐺
163157, 159, 162nf3an 1993 . . . . . . . . 9 𝑡(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙𝑓:(1...𝑚)–1-1-onto→ran 𝐺)
164142, 163nfan 1990 . . . . . . . 8 𝑡(𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙𝑓:(1...𝑚)–1-1-onto→ran 𝐺))
165141, 164nfan 1990 . . . . . . 7 𝑡(𝜑 ∧ (𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙𝑓:(1...𝑚)–1-1-onto→ran 𝐺)))
166 nfv 2005 . . . . . . . . 9 𝑤 𝑚 ∈ ℕ
167 nfcv 2959 . . . . . . . . . . 11 𝑤𝑔
168167, 11nffn 6208 . . . . . . . . . 10 𝑤 𝑔 Fn ran 𝐺
169 nfv 2005 . . . . . . . . . . 11 𝑤(𝑔𝑙) ∈ 𝑙
17011, 169nfral 3144 . . . . . . . . . 10 𝑤𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙
171 nfcv 2959 . . . . . . . . . . 11 𝑤𝑓
172 nfcv 2959 . . . . . . . . . . 11 𝑤(1...𝑚)
173171, 172, 11nff1o 6361 . . . . . . . . . 10 𝑤 𝑓:(1...𝑚)–1-1-onto→ran 𝐺
174168, 170, 173nf3an 1993 . . . . . . . . 9 𝑤(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙𝑓:(1...𝑚)–1-1-onto→ran 𝐺)
175166, 174nfan 1990 . . . . . . . 8 𝑤(𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙𝑓:(1...𝑚)–1-1-onto→ran 𝐺))
1768, 175nfan 1990 . . . . . . 7 𝑤(𝜑 ∧ (𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙𝑓:(1...𝑚)–1-1-onto→ran 𝐺)))
1772, 130, 131, 132, 135, 138, 139, 140, 165, 176, 84stoweidlem27 40741 . . . . . 6 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙𝑓:(1...𝑚)–1-1-onto→ran 𝐺))) → ∃𝑞(𝑚 ∈ ℕ ∧ (𝑞:(1...𝑚)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇𝑈)∃𝑖 ∈ (1...𝑚)0 < ((𝑞𝑖)‘𝑡))))
178177ex 399 . . . . 5 (𝜑 → ((𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙𝑓:(1...𝑚)–1-1-onto→ran 𝐺)) → ∃𝑞(𝑚 ∈ ℕ ∧ (𝑞:(1...𝑚)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇𝑈)∃𝑖 ∈ (1...𝑚)0 < ((𝑞𝑖)‘𝑡)))))
1791782eximdv 2010 . . . 4 (𝜑 → (∃𝑔𝑓(𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙𝑓:(1...𝑚)–1-1-onto→ran 𝐺)) → ∃𝑔𝑓𝑞(𝑚 ∈ ℕ ∧ (𝑞:(1...𝑚)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇𝑈)∃𝑖 ∈ (1...𝑚)0 < ((𝑞𝑖)‘𝑡)))))
180179eximdv 2008 . . 3 (𝜑 → (∃𝑚𝑔𝑓(𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙𝑓:(1...𝑚)–1-1-onto→ran 𝐺)) → ∃𝑚𝑔𝑓𝑞(𝑚 ∈ ℕ ∧ (𝑞:(1...𝑚)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇𝑈)∃𝑖 ∈ (1...𝑚)0 < ((𝑞𝑖)‘𝑡)))))
181129, 180mpd 15 . 2 (𝜑 → ∃𝑚𝑔𝑓𝑞(𝑚 ∈ ℕ ∧ (𝑞:(1...𝑚)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇𝑈)∃𝑖 ∈ (1...𝑚)0 < ((𝑞𝑖)‘𝑡))))
182 id 22 . . . 4 (∃𝑞(𝑚 ∈ ℕ ∧ (𝑞:(1...𝑚)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇𝑈)∃𝑖 ∈ (1...𝑚)0 < ((𝑞𝑖)‘𝑡))) → ∃𝑞(𝑚 ∈ ℕ ∧ (𝑞:(1...𝑚)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇𝑈)∃𝑖 ∈ (1...𝑚)0 < ((𝑞𝑖)‘𝑡))))
183182exlimivv 2023 . . 3 (∃𝑔𝑓𝑞(𝑚 ∈ ℕ ∧ (𝑞:(1...𝑚)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇𝑈)∃𝑖 ∈ (1...𝑚)0 < ((𝑞𝑖)‘𝑡))) → ∃𝑞(𝑚 ∈ ℕ ∧ (𝑞:(1...𝑚)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇𝑈)∃𝑖 ∈ (1...𝑚)0 < ((𝑞𝑖)‘𝑡))))
184183eximi 1919 . 2 (∃𝑚𝑔𝑓𝑞(𝑚 ∈ ℕ ∧ (𝑞:(1...𝑚)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇𝑈)∃𝑖 ∈ (1...𝑚)0 < ((𝑞𝑖)‘𝑡))) → ∃𝑚𝑞(𝑚 ∈ ℕ ∧ (𝑞:(1...𝑚)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇𝑈)∃𝑖 ∈ (1...𝑚)0 < ((𝑞𝑖)‘𝑡))))
185181, 184syl 17 1 (𝜑 → ∃𝑚𝑞(𝑚 ∈ ℕ ∧ (𝑞:(1...𝑚)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇𝑈)∃𝑖 ∈ (1...𝑚)0 < ((𝑞𝑖)‘𝑡))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 865  w3a 1100  wal 1635   = wceq 1637  wex 1859  wnf 1863  wcel 2157  wne 2989  wral 3107  wrex 3108  {crab 3111  Vcvv 3402  cdif 3777  wss 3780  c0 4127   cuni 4641   class class class wbr 4855  cmpt 4934  dom cdm 5324  ran crn 5325   Fn wfn 6106  wf 6107  1-1-ontowf1o 6110  cfv 6111  (class class class)co 6884  Fincfn 8202  0cc0 10231  1c1 10232   < clt 10369  cle 10370  cn 11315  ...cfz 12569  chash 13357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-rep 4977  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109  ax-un 7189  ax-cnex 10287  ax-resscn 10288  ax-1cn 10289  ax-icn 10290  ax-addcl 10291  ax-addrcl 10292  ax-mulcl 10293  ax-mulrcl 10294  ax-mulcom 10295  ax-addass 10296  ax-mulass 10297  ax-distr 10298  ax-i2m1 10299  ax-1ne0 10300  ax-1rid 10301  ax-rnegex 10302  ax-rrecex 10303  ax-cnre 10304  ax-pre-lttri 10305  ax-pre-lttrn 10306  ax-pre-ltadd 10307  ax-pre-mulgt0 10308
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-pss 3796  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-tp 4386  df-op 4388  df-uni 4642  df-int 4681  df-iun 4725  df-br 4856  df-opab 4918  df-mpt 4935  df-tr 4958  df-id 5232  df-eprel 5237  df-po 5245  df-so 5246  df-fr 5283  df-we 5285  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-pred 5907  df-ord 5953  df-on 5954  df-lim 5955  df-suc 5956  df-iota 6074  df-fun 6113  df-fn 6114  df-f 6115  df-f1 6116  df-fo 6117  df-f1o 6118  df-fv 6119  df-riota 6845  df-ov 6887  df-oprab 6888  df-mpt2 6889  df-om 7306  df-1st 7408  df-2nd 7409  df-wrecs 7652  df-recs 7714  df-rdg 7752  df-1o 7806  df-oadd 7810  df-er 7989  df-en 8203  df-dom 8204  df-sdom 8205  df-fin 8206  df-card 9058  df-pnf 10371  df-mnf 10372  df-xr 10373  df-ltxr 10374  df-le 10375  df-sub 10563  df-neg 10564  df-nn 11316  df-n0 11580  df-z 11664  df-uz 11925  df-fz 12570  df-hash 13358
This theorem is referenced by:  stoweidlem53  40767
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