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Theorem stoweidlem35 44266
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 (𝑞𝑖) is used to represent p_(ti) 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 43378 . . . . . . . . . 10 (𝑋 ∈ Fin → ran 𝐺 ∈ Fin)
41, 3syl 17 . . . . . . . . 9 (𝜑 → ran 𝐺 ∈ Fin)
5 fnchoice 43224 . . . . . . . . . . 11 (ran 𝐺 ∈ Fin → ∃𝑔(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔𝑙) ∈ 𝑙)))
65adantl 482 . . . . . . . . . 10 ((𝜑 ∧ ran 𝐺 ∈ Fin) → ∃𝑔(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔𝑙) ∈ 𝑙)))
7 simprl 769 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔𝑙) ∈ 𝑙))) → 𝑔 Fn ran 𝐺)
8 stoweidlem35.2 . . . . . . . . . . . . . . . . . . . . 21 𝑤𝜑
9 nfmpt1 5213 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑤(𝑤𝑋 ↦ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})
102, 9nfcxfr 2905 . . . . . . . . . . . . . . . . . . . . . . 23 𝑤𝐺
1110nfrn 5907 . . . . . . . . . . . . . . . . . . . . . 22 𝑤ran 𝐺
1211nfcri 2894 . . . . . . . . . . . . . . . . . . . . 21 𝑤 𝑘 ∈ ran 𝐺
138, 12nfan 1902 . . . . . . . . . . . . . . . . . . . 20 𝑤(𝜑𝑘 ∈ ran 𝐺)
14 stoweidlem35.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑𝑋𝑊)
1514sselda 3944 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑤𝑋) → 𝑤𝑊)
16 stoweidlem35.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑊 = {𝑤𝐽 ∣ ∃𝑄 𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}
1715, 16eleqtrdi 2848 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑤𝑋) → 𝑤 ∈ {𝑤𝐽 ∣ ∃𝑄 𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})
18 rabid 3427 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑤 ∈ {𝑤𝐽 ∣ ∃𝑄 𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} ↔ (𝑤𝐽 ∧ ∃𝑄 𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}))
1917, 18sylib 217 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑤𝑋) → (𝑤𝐽 ∧ ∃𝑄 𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}))
2019simprd 496 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑤𝑋) → ∃𝑄 𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)})
21 df-rex 3074 . . . . . . . . . . . . . . . . . . . . . . . . 25 (∃𝑄 𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)} ↔ ∃(𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}))
2220, 21sylib 217 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑤𝑋) → ∃(𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}))
23 rabid 3427 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( ∈ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} ↔ (𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}))
2423exbii 1850 . . . . . . . . . . . . . . . . . . . . . . . 24 (∃ ∈ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} ↔ ∃(𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}))
2522, 24sylibr 233 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑤𝑋) → ∃ ∈ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})
2625adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑤𝑋) ∧ 𝑘 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}) → ∃ ∈ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})
27 stoweidlem35.3 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝜑
28 nfv 1917 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑤𝑋
2927, 28nfan 1902 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝑤𝑋)
30 nfrab1 3426 . . . . . . . . . . . . . . . . . . . . . . . . 25 {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}
3130nfeq2 2924 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}
3229, 31nfan 1902 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑤𝑋) ∧ 𝑘 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})
33 eleq2 2826 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} → (𝑘 ∈ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}))
3433biimprd 247 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} → ( ∈ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} → 𝑘))
3534adantl 482 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑤𝑋) ∧ 𝑘 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}) → ( ∈ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} → 𝑘))
3632, 35eximd 2209 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑤𝑋) ∧ 𝑘 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}) → (∃ ∈ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} → ∃ 𝑘))
3726, 36mpd 15 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑤𝑋) ∧ 𝑘 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}) → ∃ 𝑘)
3837adantllr 717 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑘 ∈ ran 𝐺) ∧ 𝑤𝑋) ∧ 𝑘 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}) → ∃ 𝑘)
392elrnmpt 5911 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ ran 𝐺 → (𝑘 ∈ ran 𝐺 ↔ ∃𝑤𝑋 𝑘 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}))
4039ibi 266 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ ran 𝐺 → ∃𝑤𝑋 𝑘 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})
4140adantl 482 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘 ∈ ran 𝐺) → ∃𝑤𝑋 𝑘 = {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})
4213, 38, 41r19.29af 3251 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ ran 𝐺) → ∃ 𝑘)
43 n0 4306 . . . . . . . . . . . . . . . . . . 19 (𝑘 ≠ ∅ ↔ ∃ 𝑘)
4442, 43sylibr 233 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ ran 𝐺) → 𝑘 ≠ ∅)
4544adantlr 713 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔𝑙) ∈ 𝑙))) ∧ 𝑘 ∈ ran 𝐺) → 𝑘 ≠ ∅)
46 simplrr 776 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔𝑙) ∈ 𝑙))) ∧ 𝑘 ∈ ran 𝐺) → ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔𝑙) ∈ 𝑙))
47 neeq1 3006 . . . . . . . . . . . . . . . . . . . 20 (𝑙 = 𝑘 → (𝑙 ≠ ∅ ↔ 𝑘 ≠ ∅))
48 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . 22 (𝑙 = 𝑘 → (𝑔𝑙) = (𝑔𝑘))
4948eleq1d 2822 . . . . . . . . . . . . . . . . . . . . 21 (𝑙 = 𝑘 → ((𝑔𝑙) ∈ 𝑙 ↔ (𝑔𝑘) ∈ 𝑙))
50 eleq2 2826 . . . . . . . . . . . . . . . . . . . . 21 (𝑙 = 𝑘 → ((𝑔𝑘) ∈ 𝑙 ↔ (𝑔𝑘) ∈ 𝑘))
5149, 50bitrd 278 . . . . . . . . . . . . . . . . . . . 20 (𝑙 = 𝑘 → ((𝑔𝑙) ∈ 𝑙 ↔ (𝑔𝑘) ∈ 𝑘))
5247, 51imbi12d 344 . . . . . . . . . . . . . . . . . . 19 (𝑙 = 𝑘 → ((𝑙 ≠ ∅ → (𝑔𝑙) ∈ 𝑙) ↔ (𝑘 ≠ ∅ → (𝑔𝑘) ∈ 𝑘)))
5352rspccva 3580 . . . . . . . . . . . . . . . . . 18 ((∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔𝑙) ∈ 𝑙) ∧ 𝑘 ∈ ran 𝐺) → (𝑘 ≠ ∅ → (𝑔𝑘) ∈ 𝑘))
5446, 53sylancom 588 . . . . . . . . . . . . . . . . 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 3143 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔𝑙) ∈ 𝑙))) → ∀𝑘 ∈ ran 𝐺(𝑔𝑘) ∈ 𝑘)
57 fveq2 6842 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑙 → (𝑔𝑘) = (𝑔𝑙))
5857eleq1d 2822 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑙 → ((𝑔𝑘) ∈ 𝑘 ↔ (𝑔𝑙) ∈ 𝑘))
59 eleq2 2826 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑙 → ((𝑔𝑙) ∈ 𝑘 ↔ (𝑔𝑙) ∈ 𝑙))
6058, 59bitrd 278 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑙 → ((𝑔𝑘) ∈ 𝑘 ↔ (𝑔𝑙) ∈ 𝑙))
6160cbvralvw 3225 . . . . . . . . . . . . . . 15 (∀𝑘 ∈ ran 𝐺(𝑔𝑘) ∈ 𝑘 ↔ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙)
6256, 61sylib 217 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔𝑙) ∈ 𝑙))) → ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙)
637, 62jca 512 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔𝑙) ∈ 𝑙))) → (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙))
6463ex 413 . . . . . . . . . . . 12 (𝜑 → ((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑙 ≠ ∅ → (𝑔𝑙) ∈ 𝑙)) → (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙)))
6564adantr 481 . . . . . . . . . . 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 1920 . . . . . . . . . 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 685 . . . . . . . 8 (𝜑 → ∃𝑔(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙))
6968ralrimivw 3147 . . . . . . 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 4360 . . . . . . . . . . . . 13 (((𝑇𝑈) ⊆ 𝑋 ∧ (𝑇𝑈) ≠ ∅) → 𝑋 ≠ ∅)
7370, 71, 72syl2anc 584 . . . . . . . . . . . 12 (𝜑 𝑋 ≠ ∅)
7473neneqd 2948 . . . . . . . . . . 11 (𝜑 → ¬ 𝑋 = ∅)
75 unieq 4876 . . . . . . . . . . . 12 (𝑋 = ∅ → 𝑋 = ∅)
76 uni0 4896 . . . . . . . . . . . 12 ∅ = ∅
7775, 76eqtrdi 2792 . . . . . . . . . . 11 (𝑋 = ∅ → 𝑋 = ∅)
7874, 77nsyl 140 . . . . . . . . . 10 (𝜑 → ¬ 𝑋 = ∅)
79 dm0rn0 5880 . . . . . . . . . . 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 5290 . . . . . . . . . . . . . . . . 17 (𝜑𝑄 ∈ V)
83 nfrab1 3426 . . . . . . . . . . . . . . . . . . 19 {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))}
8480, 83nfcxfr 2905 . . . . . . . . . . . . . . . . . 18 𝑄
8584rabexgf 43219 . . . . . . . . . . . . . . . . 17 (𝑄 ∈ V → {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} ∈ V)
8682, 85syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} ∈ V)
8786adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑤𝑋) → {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}} ∈ V)
888, 87, 2fmptdf 7065 . . . . . . . . . . . . . 14 (𝜑𝐺:𝑋⟶V)
89 dffn2 6670 . . . . . . . . . . . . . 14 (𝐺 Fn 𝑋𝐺:𝑋⟶V)
9088, 89sylibr 233 . . . . . . . . . . . . 13 (𝜑𝐺 Fn 𝑋)
9190fndmd 6607 . . . . . . . . . . . 12 (𝜑 → dom 𝐺 = 𝑋)
9291eqeq1d 2738 . . . . . . . . . . 11 (𝜑 → (dom 𝐺 = ∅ ↔ 𝑋 = ∅))
9379, 92bitr3id 284 . . . . . . . . . 10 (𝜑 → (ran 𝐺 = ∅ ↔ 𝑋 = ∅))
9478, 93mtbird 324 . . . . . . . . 9 (𝜑 → ¬ ran 𝐺 = ∅)
95 fz1f1o 15595 . . . . . . . . . . 11 (ran 𝐺 ∈ Fin → (ran 𝐺 = ∅ ∨ ((♯‘ran 𝐺) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘ran 𝐺))–1-1-onto→ran 𝐺)))
964, 95syl 17 . . . . . . . . . 10 (𝜑 → (ran 𝐺 = ∅ ∨ ((♯‘ran 𝐺) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘ran 𝐺))–1-1-onto→ran 𝐺)))
9796ord 862 . . . . . . . . 9 (𝜑 → (¬ ran 𝐺 = ∅ → ((♯‘ran 𝐺) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘ran 𝐺))–1-1-onto→ran 𝐺)))
9894, 97mpd 15 . . . . . . . 8 (𝜑 → ((♯‘ran 𝐺) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘ran 𝐺))–1-1-onto→ran 𝐺))
99 oveq2 7365 . . . . . . . . . . 11 (𝑚 = (♯‘ran 𝐺) → (1...𝑚) = (1...(♯‘ran 𝐺)))
10099f1oeq2d 6780 . . . . . . . . . 10 (𝑚 = (♯‘ran 𝐺) → (𝑓:(1...𝑚)–1-1-onto→ran 𝐺𝑓:(1...(♯‘ran 𝐺))–1-1-onto→ran 𝐺))
101100exbidv 1924 . . . . . . . . 9 (𝑚 = (♯‘ran 𝐺) → (∃𝑓 𝑓:(1...𝑚)–1-1-onto→ran 𝐺 ↔ ∃𝑓 𝑓:(1...(♯‘ran 𝐺))–1-1-onto→ran 𝐺))
102101rspcev 3581 . . . . . . . 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)
10398, 102syl 17 . . . . . . 7 (𝜑 → ∃𝑚 ∈ ℕ ∃𝑓 𝑓:(1...𝑚)–1-1-onto→ran 𝐺)
104 r19.29 3117 . . . . . . 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))
10569, 103, 104syl2anc 584 . . . . . 6 (𝜑 → ∃𝑚 ∈ ℕ (∃𝑔(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙) ∧ ∃𝑓 𝑓:(1...𝑚)–1-1-onto→ran 𝐺))
106 exdistrv 1959 . . . . . . . . 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))
107106biimpri 227 . . . . . . . 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))
108107a1i 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)))
109108reximdv 3167 . . . . . 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)))
110105, 109mpd 15 . . . . 5 (𝜑 → ∃𝑚 ∈ ℕ ∃𝑔𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙) ∧ 𝑓:(1...𝑚)–1-1-onto→ran 𝐺))
111 df-rex 3074 . . . . 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)))
112110, 111sylib 217 . . . 4 (𝜑 → ∃𝑚(𝑚 ∈ ℕ ∧ ∃𝑔𝑓((𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 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 ∧ ∃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)))
115113, 114sylan 580 . . . . . . . 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)))
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(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)))
118116, 117sylan 580 . . . . . . . . 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)))
119118eximi 1837 . . . . . . . 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)))
120115, 119syl 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)))
121 df-3an 1089 . . . . . . . . 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))
122121anbi2i 623 . . . . . . . 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)))
1231222exbii 1851 . . . . . . 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)))
124120, 123sylibr 233 . . . . . 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)))
125124a1i 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))))
126125eximdv 1920 . . . 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))))
127112, 126mpd 15 . . 3 (𝜑 → ∃𝑚𝑔𝑓(𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙𝑓:(1...𝑚)–1-1-onto→ran 𝐺)))
12882adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙𝑓:(1...𝑚)–1-1-onto→ran 𝐺))) → 𝑄 ∈ V)
129 simprl 769 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙𝑓:(1...𝑚)–1-1-onto→ran 𝐺))) → 𝑚 ∈ ℕ)
130 simprr1 1221 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙𝑓:(1...𝑚)–1-1-onto→ran 𝐺))) → 𝑔 Fn ran 𝐺)
131 elex 3463 . . . . . . . . 9 (ran 𝐺 ∈ Fin → ran 𝐺 ∈ V)
1324, 131syl 17 . . . . . . . 8 (𝜑 → ran 𝐺 ∈ V)
133132adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙𝑓:(1...𝑚)–1-1-onto→ran 𝐺))) → ran 𝐺 ∈ V)
134 simprr2 1222 . . . . . . . 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)
13551rspccva 3580 . . . . . . . 8 ((∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙𝑘 ∈ ran 𝐺) → (𝑔𝑘) ∈ 𝑘)
136134, 135sylan 580 . . . . . . 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)
137 simprr3 1223 . . . . . . 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)
13870adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙𝑓:(1...𝑚)–1-1-onto→ran 𝐺))) → (𝑇𝑈) ⊆ 𝑋)
139 stoweidlem35.1 . . . . . . . 8 𝑡𝜑
140 nfv 1917 . . . . . . . . 9 𝑡 𝑚 ∈ ℕ
141 nfcv 2907 . . . . . . . . . . 11 𝑡𝑔
142 nfcv 2907 . . . . . . . . . . . . . 14 𝑡𝑋
143 nfrab1 3426 . . . . . . . . . . . . . . . 16 𝑡{𝑡𝑇 ∣ 0 < (𝑡)}
144143nfeq2 2924 . . . . . . . . . . . . . . 15 𝑡 𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}
145 nfv 1917 . . . . . . . . . . . . . . . . . 18 𝑡(𝑍) = 0
146 nfra1 3267 . . . . . . . . . . . . . . . . . 18 𝑡𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)
147145, 146nfan 1902 . . . . . . . . . . . . . . . . 17 𝑡((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))
148 nfcv 2907 . . . . . . . . . . . . . . . . 17 𝑡𝐴
149147, 148nfrabw 3440 . . . . . . . . . . . . . . . 16 𝑡{𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))}
15080, 149nfcxfr 2905 . . . . . . . . . . . . . . 15 𝑡𝑄
151144, 150nfrabw 3440 . . . . . . . . . . . . . 14 𝑡{𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}
152142, 151nfmpt 5212 . . . . . . . . . . . . 13 𝑡(𝑤𝑋 ↦ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})
1532, 152nfcxfr 2905 . . . . . . . . . . . 12 𝑡𝐺
154153nfrn 5907 . . . . . . . . . . 11 𝑡ran 𝐺
155141, 154nffn 6601 . . . . . . . . . 10 𝑡 𝑔 Fn ran 𝐺
156 nfv 1917 . . . . . . . . . . 11 𝑡(𝑔𝑙) ∈ 𝑙
157154, 156nfralw 3294 . . . . . . . . . 10 𝑡𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙
158 nfcv 2907 . . . . . . . . . . 11 𝑡𝑓
159 nfcv 2907 . . . . . . . . . . 11 𝑡(1...𝑚)
160158, 159, 154nff1o 6782 . . . . . . . . . 10 𝑡 𝑓:(1...𝑚)–1-1-onto→ran 𝐺
161155, 157, 160nf3an 1904 . . . . . . . . 9 𝑡(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙𝑓:(1...𝑚)–1-1-onto→ran 𝐺)
162140, 161nfan 1902 . . . . . . . 8 𝑡(𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙𝑓:(1...𝑚)–1-1-onto→ran 𝐺))
163139, 162nfan 1902 . . . . . . 7 𝑡(𝜑 ∧ (𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙𝑓:(1...𝑚)–1-1-onto→ran 𝐺)))
164 nfv 1917 . . . . . . . . 9 𝑤 𝑚 ∈ ℕ
165 nfcv 2907 . . . . . . . . . . 11 𝑤𝑔
166165, 11nffn 6601 . . . . . . . . . 10 𝑤 𝑔 Fn ran 𝐺
167 nfv 1917 . . . . . . . . . . 11 𝑤(𝑔𝑙) ∈ 𝑙
16811, 167nfralw 3294 . . . . . . . . . 10 𝑤𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙
169 nfcv 2907 . . . . . . . . . . 11 𝑤𝑓
170 nfcv 2907 . . . . . . . . . . 11 𝑤(1...𝑚)
171169, 170, 11nff1o 6782 . . . . . . . . . 10 𝑤 𝑓:(1...𝑚)–1-1-onto→ran 𝐺
172166, 168, 171nf3an 1904 . . . . . . . . 9 𝑤(𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙𝑓:(1...𝑚)–1-1-onto→ran 𝐺)
173164, 172nfan 1902 . . . . . . . 8 𝑤(𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙𝑓:(1...𝑚)–1-1-onto→ran 𝐺))
1748, 173nfan 1902 . . . . . . 7 𝑤(𝜑 ∧ (𝑚 ∈ ℕ ∧ (𝑔 Fn ran 𝐺 ∧ ∀𝑙 ∈ ran 𝐺(𝑔𝑙) ∈ 𝑙𝑓:(1...𝑚)–1-1-onto→ran 𝐺)))
1752, 128, 129, 130, 133, 136, 137, 138, 163, 174, 84stoweidlem27 44258 . . . . . 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))))
176175ex 413 . . . . 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)))))
1771762eximdv 1922 . . . 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)))))
178177eximdv 1920 . . 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)))))
179127, 178mpd 15 . 2 (𝜑 → ∃𝑚𝑔𝑓𝑞(𝑚 ∈ ℕ ∧ (𝑞:(1...𝑚)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇𝑈)∃𝑖 ∈ (1...𝑚)0 < ((𝑞𝑖)‘𝑡))))
180 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))))
181180exlimivv 1935 . . 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))))
182181eximi 1837 . 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))))
183179, 182syl 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  wa 396  wo 845  w3a 1087  wal 1539   = wceq 1541  wex 1781  wnf 1785  wcel 2106  wne 2943  wral 3064  wrex 3073  {crab 3407  Vcvv 3445  cdif 3907  wss 3910  c0 4282   cuni 4865   class class class wbr 5105  cmpt 5188  dom cdm 5633  ran crn 5634   Fn wfn 6491  wf 6492  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  Fincfn 8883  0cc0 11051  1c1 11052   < clt 11189  cle 11190  cn 12153  ...cfz 13424  chash 14230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-hash 14231
This theorem is referenced by:  stoweidlem53  44284
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