Metamath Proof Explorer


Theorem stoweidlem27

Description: This lemma is used to prove the existence of a function p as in Lemma 1 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)

Ref Expression
Hypotheses stoweidlem27.1 𝐺 = ( 𝑤𝑋 ↦ { 𝑄𝑤 = { 𝑡𝑇 ∣ 0 < ( 𝑡 ) } } )
stoweidlem27.2 ( 𝜑𝑄 ∈ V )
stoweidlem27.3 ( 𝜑𝑀 ∈ ℕ )
stoweidlem27.4 ( 𝜑𝑌 Fn ran 𝐺 )
stoweidlem27.5 ( 𝜑 → ran 𝐺 ∈ V )
stoweidlem27.6 ( ( 𝜑𝑙 ∈ ran 𝐺 ) → ( 𝑌𝑙 ) ∈ 𝑙 )
stoweidlem27.7 ( 𝜑𝐹 : ( 1 ... 𝑀 ) –1-1-onto→ ran 𝐺 )
stoweidlem27.8 ( 𝜑 → ( 𝑇𝑈 ) ⊆ 𝑋 )
stoweidlem27.9 𝑡 𝜑
stoweidlem27.10 𝑤 𝜑
stoweidlem27.11 𝑄
Assertion stoweidlem27 ( 𝜑 → ∃ 𝑞 ( 𝑀 ∈ ℕ ∧ ( 𝑞 : ( 1 ... 𝑀 ) ⟶ 𝑄 ∧ ∀ 𝑡 ∈ ( 𝑇𝑈 ) ∃ 𝑖 ∈ ( 1 ... 𝑀 ) 0 < ( ( 𝑞𝑖 ) ‘ 𝑡 ) ) ) )

Proof

Step Hyp Ref Expression
1 stoweidlem27.1 𝐺 = ( 𝑤𝑋 ↦ { 𝑄𝑤 = { 𝑡𝑇 ∣ 0 < ( 𝑡 ) } } )
2 stoweidlem27.2 ( 𝜑𝑄 ∈ V )
3 stoweidlem27.3 ( 𝜑𝑀 ∈ ℕ )
4 stoweidlem27.4 ( 𝜑𝑌 Fn ran 𝐺 )
5 stoweidlem27.5 ( 𝜑 → ran 𝐺 ∈ V )
6 stoweidlem27.6 ( ( 𝜑𝑙 ∈ ran 𝐺 ) → ( 𝑌𝑙 ) ∈ 𝑙 )
7 stoweidlem27.7 ( 𝜑𝐹 : ( 1 ... 𝑀 ) –1-1-onto→ ran 𝐺 )
8 stoweidlem27.8 ( 𝜑 → ( 𝑇𝑈 ) ⊆ 𝑋 )
9 stoweidlem27.9 𝑡 𝜑
10 stoweidlem27.10 𝑤 𝜑
11 stoweidlem27.11 𝑄
12 fnex ( ( 𝑌 Fn ran 𝐺 ∧ ran 𝐺 ∈ V ) → 𝑌 ∈ V )
13 4 5 12 syl2anc ( 𝜑𝑌 ∈ V )
14 f1ofn ( 𝐹 : ( 1 ... 𝑀 ) –1-1-onto→ ran 𝐺𝐹 Fn ( 1 ... 𝑀 ) )
15 7 14 syl ( 𝜑𝐹 Fn ( 1 ... 𝑀 ) )
16 ovex ( 1 ... 𝑀 ) ∈ V
17 fnex ( ( 𝐹 Fn ( 1 ... 𝑀 ) ∧ ( 1 ... 𝑀 ) ∈ V ) → 𝐹 ∈ V )
18 15 16 17 sylancl ( 𝜑𝐹 ∈ V )
19 coexg ( ( 𝑌 ∈ V ∧ 𝐹 ∈ V ) → ( 𝑌𝐹 ) ∈ V )
20 13 18 19 syl2anc ( 𝜑 → ( 𝑌𝐹 ) ∈ V )
21 f1of ( 𝐹 : ( 1 ... 𝑀 ) –1-1-onto→ ran 𝐺𝐹 : ( 1 ... 𝑀 ) ⟶ ran 𝐺 )
22 7 21 syl ( 𝜑𝐹 : ( 1 ... 𝑀 ) ⟶ ran 𝐺 )
23 fnfco ( ( 𝑌 Fn ran 𝐺𝐹 : ( 1 ... 𝑀 ) ⟶ ran 𝐺 ) → ( 𝑌𝐹 ) Fn ( 1 ... 𝑀 ) )
24 4 22 23 syl2anc ( 𝜑 → ( 𝑌𝐹 ) Fn ( 1 ... 𝑀 ) )
25 rncoss ran ( 𝑌𝐹 ) ⊆ ran 𝑌
26 fvelrnb ( 𝑌 Fn ran 𝐺 → ( 𝑘 ∈ ran 𝑌 ↔ ∃ 𝑙 ∈ ran 𝐺 ( 𝑌𝑙 ) = 𝑘 ) )
27 4 26 syl ( 𝜑 → ( 𝑘 ∈ ran 𝑌 ↔ ∃ 𝑙 ∈ ran 𝐺 ( 𝑌𝑙 ) = 𝑘 ) )
28 27 biimpa ( ( 𝜑𝑘 ∈ ran 𝑌 ) → ∃ 𝑙 ∈ ran 𝐺 ( 𝑌𝑙 ) = 𝑘 )
29 nfmpt1 𝑤 ( 𝑤𝑋 ↦ { 𝑄𝑤 = { 𝑡𝑇 ∣ 0 < ( 𝑡 ) } } )
30 1 29 nfcxfr 𝑤 𝐺
31 30 nfrn 𝑤 ran 𝐺
32 31 nfcri 𝑤 𝑙 ∈ ran 𝐺
33 10 32 nfan 𝑤 ( 𝜑𝑙 ∈ ran 𝐺 )
34 6 ad2antrr ( ( ( ( 𝜑𝑙 ∈ ran 𝐺 ) ∧ 𝑤𝑋 ) ∧ 𝑙 = { 𝑄𝑤 = { 𝑡𝑇 ∣ 0 < ( 𝑡 ) } } ) → ( 𝑌𝑙 ) ∈ 𝑙 )
35 simpr ( ( ( ( 𝜑𝑙 ∈ ran 𝐺 ) ∧ 𝑤𝑋 ) ∧ 𝑙 = { 𝑄𝑤 = { 𝑡𝑇 ∣ 0 < ( 𝑡 ) } } ) → 𝑙 = { 𝑄𝑤 = { 𝑡𝑇 ∣ 0 < ( 𝑡 ) } } )
36 34 35 eleqtrd ( ( ( ( 𝜑𝑙 ∈ ran 𝐺 ) ∧ 𝑤𝑋 ) ∧ 𝑙 = { 𝑄𝑤 = { 𝑡𝑇 ∣ 0 < ( 𝑡 ) } } ) → ( 𝑌𝑙 ) ∈ { 𝑄𝑤 = { 𝑡𝑇 ∣ 0 < ( 𝑡 ) } } )
37 nfcv ( 𝑌𝑙 )
38 nfv 𝑤 = { 𝑡𝑇 ∣ 0 < ( ( 𝑌𝑙 ) ‘ 𝑡 ) }
39 fveq1 ( = ( 𝑌𝑙 ) → ( 𝑡 ) = ( ( 𝑌𝑙 ) ‘ 𝑡 ) )
40 39 breq2d ( = ( 𝑌𝑙 ) → ( 0 < ( 𝑡 ) ↔ 0 < ( ( 𝑌𝑙 ) ‘ 𝑡 ) ) )
41 40 rabbidv ( = ( 𝑌𝑙 ) → { 𝑡𝑇 ∣ 0 < ( 𝑡 ) } = { 𝑡𝑇 ∣ 0 < ( ( 𝑌𝑙 ) ‘ 𝑡 ) } )
42 41 eqeq2d ( = ( 𝑌𝑙 ) → ( 𝑤 = { 𝑡𝑇 ∣ 0 < ( 𝑡 ) } ↔ 𝑤 = { 𝑡𝑇 ∣ 0 < ( ( 𝑌𝑙 ) ‘ 𝑡 ) } ) )
43 37 11 38 42 elrabf ( ( 𝑌𝑙 ) ∈ { 𝑄𝑤 = { 𝑡𝑇 ∣ 0 < ( 𝑡 ) } } ↔ ( ( 𝑌𝑙 ) ∈ 𝑄𝑤 = { 𝑡𝑇 ∣ 0 < ( ( 𝑌𝑙 ) ‘ 𝑡 ) } ) )
44 36 43 sylib ( ( ( ( 𝜑𝑙 ∈ ran 𝐺 ) ∧ 𝑤𝑋 ) ∧ 𝑙 = { 𝑄𝑤 = { 𝑡𝑇 ∣ 0 < ( 𝑡 ) } } ) → ( ( 𝑌𝑙 ) ∈ 𝑄𝑤 = { 𝑡𝑇 ∣ 0 < ( ( 𝑌𝑙 ) ‘ 𝑡 ) } ) )
45 44 simpld ( ( ( ( 𝜑𝑙 ∈ ran 𝐺 ) ∧ 𝑤𝑋 ) ∧ 𝑙 = { 𝑄𝑤 = { 𝑡𝑇 ∣ 0 < ( 𝑡 ) } } ) → ( 𝑌𝑙 ) ∈ 𝑄 )
46 simpr ( ( 𝜑𝑙 ∈ ran 𝐺 ) → 𝑙 ∈ ran 𝐺 )
47 1 elrnmpt ( 𝑙 ∈ ran 𝐺 → ( 𝑙 ∈ ran 𝐺 ↔ ∃ 𝑤𝑋 𝑙 = { 𝑄𝑤 = { 𝑡𝑇 ∣ 0 < ( 𝑡 ) } } ) )
48 46 47 syl ( ( 𝜑𝑙 ∈ ran 𝐺 ) → ( 𝑙 ∈ ran 𝐺 ↔ ∃ 𝑤𝑋 𝑙 = { 𝑄𝑤 = { 𝑡𝑇 ∣ 0 < ( 𝑡 ) } } ) )
49 46 48 mpbid ( ( 𝜑𝑙 ∈ ran 𝐺 ) → ∃ 𝑤𝑋 𝑙 = { 𝑄𝑤 = { 𝑡𝑇 ∣ 0 < ( 𝑡 ) } } )
50 33 45 49 r19.29af ( ( 𝜑𝑙 ∈ ran 𝐺 ) → ( 𝑌𝑙 ) ∈ 𝑄 )
51 50 adantlr ( ( ( 𝜑𝑘 ∈ ran 𝑌 ) ∧ 𝑙 ∈ ran 𝐺 ) → ( 𝑌𝑙 ) ∈ 𝑄 )
52 eleq1 ( ( 𝑌𝑙 ) = 𝑘 → ( ( 𝑌𝑙 ) ∈ 𝑄𝑘𝑄 ) )
53 51 52 syl5ibcom ( ( ( 𝜑𝑘 ∈ ran 𝑌 ) ∧ 𝑙 ∈ ran 𝐺 ) → ( ( 𝑌𝑙 ) = 𝑘𝑘𝑄 ) )
54 53 reximdva ( ( 𝜑𝑘 ∈ ran 𝑌 ) → ( ∃ 𝑙 ∈ ran 𝐺 ( 𝑌𝑙 ) = 𝑘 → ∃ 𝑙 ∈ ran 𝐺 𝑘𝑄 ) )
55 28 54 mpd ( ( 𝜑𝑘 ∈ ran 𝑌 ) → ∃ 𝑙 ∈ ran 𝐺 𝑘𝑄 )
56 idd ( 𝑙 ∈ ran 𝐺 → ( 𝑘𝑄𝑘𝑄 ) )
57 56 a1i ( ( 𝜑𝑘 ∈ ran 𝑌 ) → ( 𝑙 ∈ ran 𝐺 → ( 𝑘𝑄𝑘𝑄 ) ) )
58 57 rexlimdv ( ( 𝜑𝑘 ∈ ran 𝑌 ) → ( ∃ 𝑙 ∈ ran 𝐺 𝑘𝑄𝑘𝑄 ) )
59 55 58 mpd ( ( 𝜑𝑘 ∈ ran 𝑌 ) → 𝑘𝑄 )
60 59 ex ( 𝜑 → ( 𝑘 ∈ ran 𝑌𝑘𝑄 ) )
61 60 ssrdv ( 𝜑 → ran 𝑌𝑄 )
62 25 61 sstrid ( 𝜑 → ran ( 𝑌𝐹 ) ⊆ 𝑄 )
63 df-f ( ( 𝑌𝐹 ) : ( 1 ... 𝑀 ) ⟶ 𝑄 ↔ ( ( 𝑌𝐹 ) Fn ( 1 ... 𝑀 ) ∧ ran ( 𝑌𝐹 ) ⊆ 𝑄 ) )
64 24 62 63 sylanbrc ( 𝜑 → ( 𝑌𝐹 ) : ( 1 ... 𝑀 ) ⟶ 𝑄 )
65 nfv 𝑤 𝑡 ∈ ( 𝑇𝑈 )
66 10 65 nfan 𝑤 ( 𝜑𝑡 ∈ ( 𝑇𝑈 ) )
67 nfv 𝑤𝑖 ∈ ( 1 ... 𝑀 ) 0 < ( ( ( 𝑌𝐹 ) ‘ 𝑖 ) ‘ 𝑡 )
68 8 sselda ( ( 𝜑𝑡 ∈ ( 𝑇𝑈 ) ) → 𝑡 𝑋 )
69 eluni ( 𝑡 𝑋 ↔ ∃ 𝑤 ( 𝑡𝑤𝑤𝑋 ) )
70 68 69 sylib ( ( 𝜑𝑡 ∈ ( 𝑇𝑈 ) ) → ∃ 𝑤 ( 𝑡𝑤𝑤𝑋 ) )
71 1 funmpt2 Fun 𝐺
72 1 dmeqi dom 𝐺 = dom ( 𝑤𝑋 ↦ { 𝑄𝑤 = { 𝑡𝑇 ∣ 0 < ( 𝑡 ) } } )
73 11 rabexgf ( 𝑄 ∈ V → { 𝑄𝑤 = { 𝑡𝑇 ∣ 0 < ( 𝑡 ) } } ∈ V )
74 2 73 syl ( 𝜑 → { 𝑄𝑤 = { 𝑡𝑇 ∣ 0 < ( 𝑡 ) } } ∈ V )
75 74 adantr ( ( 𝜑𝑤𝑋 ) → { 𝑄𝑤 = { 𝑡𝑇 ∣ 0 < ( 𝑡 ) } } ∈ V )
76 75 ex ( 𝜑 → ( 𝑤𝑋 → { 𝑄𝑤 = { 𝑡𝑇 ∣ 0 < ( 𝑡 ) } } ∈ V ) )
77 10 76 ralrimi ( 𝜑 → ∀ 𝑤𝑋 { 𝑄𝑤 = { 𝑡𝑇 ∣ 0 < ( 𝑡 ) } } ∈ V )
78 dmmptg ( ∀ 𝑤𝑋 { 𝑄𝑤 = { 𝑡𝑇 ∣ 0 < ( 𝑡 ) } } ∈ V → dom ( 𝑤𝑋 ↦ { 𝑄𝑤 = { 𝑡𝑇 ∣ 0 < ( 𝑡 ) } } ) = 𝑋 )
79 77 78 syl ( 𝜑 → dom ( 𝑤𝑋 ↦ { 𝑄𝑤 = { 𝑡𝑇 ∣ 0 < ( 𝑡 ) } } ) = 𝑋 )
80 72 79 syl5eq ( 𝜑 → dom 𝐺 = 𝑋 )
81 80 eleq2d ( 𝜑 → ( 𝑤 ∈ dom 𝐺𝑤𝑋 ) )
82 81 biimpar ( ( 𝜑𝑤𝑋 ) → 𝑤 ∈ dom 𝐺 )
83 fvelrn ( ( Fun 𝐺𝑤 ∈ dom 𝐺 ) → ( 𝐺𝑤 ) ∈ ran 𝐺 )
84 71 82 83 sylancr ( ( 𝜑𝑤𝑋 ) → ( 𝐺𝑤 ) ∈ ran 𝐺 )
85 84 adantrl ( ( 𝜑 ∧ ( 𝑡𝑤𝑤𝑋 ) ) → ( 𝐺𝑤 ) ∈ ran 𝐺 )
86 22 ad2antrr ( ( ( 𝜑 ∧ ( 𝐺𝑤 ) ∈ ran 𝐺 ) ∧ ( 𝑖 ∈ ( 1 ... 𝑀 ) ∧ ( 𝐹𝑖 ) = ( 𝐺𝑤 ) ) ) → 𝐹 : ( 1 ... 𝑀 ) ⟶ ran 𝐺 )
87 simprl ( ( ( 𝜑 ∧ ( 𝐺𝑤 ) ∈ ran 𝐺 ) ∧ ( 𝑖 ∈ ( 1 ... 𝑀 ) ∧ ( 𝐹𝑖 ) = ( 𝐺𝑤 ) ) ) → 𝑖 ∈ ( 1 ... 𝑀 ) )
88 fvco3 ( ( 𝐹 : ( 1 ... 𝑀 ) ⟶ ran 𝐺𝑖 ∈ ( 1 ... 𝑀 ) ) → ( ( 𝑌𝐹 ) ‘ 𝑖 ) = ( 𝑌 ‘ ( 𝐹𝑖 ) ) )
89 86 87 88 syl2anc ( ( ( 𝜑 ∧ ( 𝐺𝑤 ) ∈ ran 𝐺 ) ∧ ( 𝑖 ∈ ( 1 ... 𝑀 ) ∧ ( 𝐹𝑖 ) = ( 𝐺𝑤 ) ) ) → ( ( 𝑌𝐹 ) ‘ 𝑖 ) = ( 𝑌 ‘ ( 𝐹𝑖 ) ) )
90 fveq2 ( ( 𝐹𝑖 ) = ( 𝐺𝑤 ) → ( 𝑌 ‘ ( 𝐹𝑖 ) ) = ( 𝑌 ‘ ( 𝐺𝑤 ) ) )
91 90 ad2antll ( ( ( 𝜑 ∧ ( 𝐺𝑤 ) ∈ ran 𝐺 ) ∧ ( 𝑖 ∈ ( 1 ... 𝑀 ) ∧ ( 𝐹𝑖 ) = ( 𝐺𝑤 ) ) ) → ( 𝑌 ‘ ( 𝐹𝑖 ) ) = ( 𝑌 ‘ ( 𝐺𝑤 ) ) )
92 89 91 eqtrd ( ( ( 𝜑 ∧ ( 𝐺𝑤 ) ∈ ran 𝐺 ) ∧ ( 𝑖 ∈ ( 1 ... 𝑀 ) ∧ ( 𝐹𝑖 ) = ( 𝐺𝑤 ) ) ) → ( ( 𝑌𝐹 ) ‘ 𝑖 ) = ( 𝑌 ‘ ( 𝐺𝑤 ) ) )
93 eleq1 ( 𝑙 = ( 𝐺𝑤 ) → ( 𝑙 ∈ ran 𝐺 ↔ ( 𝐺𝑤 ) ∈ ran 𝐺 ) )
94 93 anbi2d ( 𝑙 = ( 𝐺𝑤 ) → ( ( 𝜑𝑙 ∈ ran 𝐺 ) ↔ ( 𝜑 ∧ ( 𝐺𝑤 ) ∈ ran 𝐺 ) ) )
95 eleq2 ( 𝑙 = ( 𝐺𝑤 ) → ( ( 𝑌𝑙 ) ∈ 𝑙 ↔ ( 𝑌𝑙 ) ∈ ( 𝐺𝑤 ) ) )
96 fveq2 ( 𝑙 = ( 𝐺𝑤 ) → ( 𝑌𝑙 ) = ( 𝑌 ‘ ( 𝐺𝑤 ) ) )
97 96 eleq1d ( 𝑙 = ( 𝐺𝑤 ) → ( ( 𝑌𝑙 ) ∈ ( 𝐺𝑤 ) ↔ ( 𝑌 ‘ ( 𝐺𝑤 ) ) ∈ ( 𝐺𝑤 ) ) )
98 95 97 bitrd ( 𝑙 = ( 𝐺𝑤 ) → ( ( 𝑌𝑙 ) ∈ 𝑙 ↔ ( 𝑌 ‘ ( 𝐺𝑤 ) ) ∈ ( 𝐺𝑤 ) ) )
99 94 98 imbi12d ( 𝑙 = ( 𝐺𝑤 ) → ( ( ( 𝜑𝑙 ∈ ran 𝐺 ) → ( 𝑌𝑙 ) ∈ 𝑙 ) ↔ ( ( 𝜑 ∧ ( 𝐺𝑤 ) ∈ ran 𝐺 ) → ( 𝑌 ‘ ( 𝐺𝑤 ) ) ∈ ( 𝐺𝑤 ) ) ) )
100 99 6 vtoclg ( ( 𝐺𝑤 ) ∈ ran 𝐺 → ( ( 𝜑 ∧ ( 𝐺𝑤 ) ∈ ran 𝐺 ) → ( 𝑌 ‘ ( 𝐺𝑤 ) ) ∈ ( 𝐺𝑤 ) ) )
101 100 anabsi7 ( ( 𝜑 ∧ ( 𝐺𝑤 ) ∈ ran 𝐺 ) → ( 𝑌 ‘ ( 𝐺𝑤 ) ) ∈ ( 𝐺𝑤 ) )
102 101 adantr ( ( ( 𝜑 ∧ ( 𝐺𝑤 ) ∈ ran 𝐺 ) ∧ ( 𝑖 ∈ ( 1 ... 𝑀 ) ∧ ( 𝐹𝑖 ) = ( 𝐺𝑤 ) ) ) → ( 𝑌 ‘ ( 𝐺𝑤 ) ) ∈ ( 𝐺𝑤 ) )
103 92 102 eqeltrd ( ( ( 𝜑 ∧ ( 𝐺𝑤 ) ∈ ran 𝐺 ) ∧ ( 𝑖 ∈ ( 1 ... 𝑀 ) ∧ ( 𝐹𝑖 ) = ( 𝐺𝑤 ) ) ) → ( ( 𝑌𝐹 ) ‘ 𝑖 ) ∈ ( 𝐺𝑤 ) )
104 f1ofo ( 𝐹 : ( 1 ... 𝑀 ) –1-1-onto→ ran 𝐺𝐹 : ( 1 ... 𝑀 ) –onto→ ran 𝐺 )
105 forn ( 𝐹 : ( 1 ... 𝑀 ) –onto→ ran 𝐺 → ran 𝐹 = ran 𝐺 )
106 7 104 105 3syl ( 𝜑 → ran 𝐹 = ran 𝐺 )
107 106 eleq2d ( 𝜑 → ( ( 𝐺𝑤 ) ∈ ran 𝐹 ↔ ( 𝐺𝑤 ) ∈ ran 𝐺 ) )
108 107 biimpar ( ( 𝜑 ∧ ( 𝐺𝑤 ) ∈ ran 𝐺 ) → ( 𝐺𝑤 ) ∈ ran 𝐹 )
109 15 adantr ( ( 𝜑 ∧ ( 𝐺𝑤 ) ∈ ran 𝐺 ) → 𝐹 Fn ( 1 ... 𝑀 ) )
110 fvelrnb ( 𝐹 Fn ( 1 ... 𝑀 ) → ( ( 𝐺𝑤 ) ∈ ran 𝐹 ↔ ∃ 𝑖 ∈ ( 1 ... 𝑀 ) ( 𝐹𝑖 ) = ( 𝐺𝑤 ) ) )
111 109 110 syl ( ( 𝜑 ∧ ( 𝐺𝑤 ) ∈ ran 𝐺 ) → ( ( 𝐺𝑤 ) ∈ ran 𝐹 ↔ ∃ 𝑖 ∈ ( 1 ... 𝑀 ) ( 𝐹𝑖 ) = ( 𝐺𝑤 ) ) )
112 108 111 mpbid ( ( 𝜑 ∧ ( 𝐺𝑤 ) ∈ ran 𝐺 ) → ∃ 𝑖 ∈ ( 1 ... 𝑀 ) ( 𝐹𝑖 ) = ( 𝐺𝑤 ) )
113 103 112 reximddv ( ( 𝜑 ∧ ( 𝐺𝑤 ) ∈ ran 𝐺 ) → ∃ 𝑖 ∈ ( 1 ... 𝑀 ) ( ( 𝑌𝐹 ) ‘ 𝑖 ) ∈ ( 𝐺𝑤 ) )
114 85 113 syldan ( ( 𝜑 ∧ ( 𝑡𝑤𝑤𝑋 ) ) → ∃ 𝑖 ∈ ( 1 ... 𝑀 ) ( ( 𝑌𝐹 ) ‘ 𝑖 ) ∈ ( 𝐺𝑤 ) )
115 simplrl ( ( ( 𝜑 ∧ ( 𝑡𝑤𝑤𝑋 ) ) ∧ ( ( 𝑌𝐹 ) ‘ 𝑖 ) ∈ ( 𝐺𝑤 ) ) → 𝑡𝑤 )
116 simpr ( ( 𝜑𝑤𝑋 ) → 𝑤𝑋 )
117 1 fvmpt2 ( ( 𝑤𝑋 ∧ { 𝑄𝑤 = { 𝑡𝑇 ∣ 0 < ( 𝑡 ) } } ∈ V ) → ( 𝐺𝑤 ) = { 𝑄𝑤 = { 𝑡𝑇 ∣ 0 < ( 𝑡 ) } } )
118 116 75 117 syl2anc ( ( 𝜑𝑤𝑋 ) → ( 𝐺𝑤 ) = { 𝑄𝑤 = { 𝑡𝑇 ∣ 0 < ( 𝑡 ) } } )
119 118 eleq2d ( ( 𝜑𝑤𝑋 ) → ( ( ( 𝑌𝐹 ) ‘ 𝑖 ) ∈ ( 𝐺𝑤 ) ↔ ( ( 𝑌𝐹 ) ‘ 𝑖 ) ∈ { 𝑄𝑤 = { 𝑡𝑇 ∣ 0 < ( 𝑡 ) } } ) )
120 119 biimpa ( ( ( 𝜑𝑤𝑋 ) ∧ ( ( 𝑌𝐹 ) ‘ 𝑖 ) ∈ ( 𝐺𝑤 ) ) → ( ( 𝑌𝐹 ) ‘ 𝑖 ) ∈ { 𝑄𝑤 = { 𝑡𝑇 ∣ 0 < ( 𝑡 ) } } )
121 120 adantlrl ( ( ( 𝜑 ∧ ( 𝑡𝑤𝑤𝑋 ) ) ∧ ( ( 𝑌𝐹 ) ‘ 𝑖 ) ∈ ( 𝐺𝑤 ) ) → ( ( 𝑌𝐹 ) ‘ 𝑖 ) ∈ { 𝑄𝑤 = { 𝑡𝑇 ∣ 0 < ( 𝑡 ) } } )
122 nfcv ( ( 𝑌𝐹 ) ‘ 𝑖 )
123 nfv 𝑤 = { 𝑡𝑇 ∣ 0 < ( ( ( 𝑌𝐹 ) ‘ 𝑖 ) ‘ 𝑡 ) }
124 fveq1 ( = ( ( 𝑌𝐹 ) ‘ 𝑖 ) → ( 𝑡 ) = ( ( ( 𝑌𝐹 ) ‘ 𝑖 ) ‘ 𝑡 ) )
125 124 breq2d ( = ( ( 𝑌𝐹 ) ‘ 𝑖 ) → ( 0 < ( 𝑡 ) ↔ 0 < ( ( ( 𝑌𝐹 ) ‘ 𝑖 ) ‘ 𝑡 ) ) )
126 125 rabbidv ( = ( ( 𝑌𝐹 ) ‘ 𝑖 ) → { 𝑡𝑇 ∣ 0 < ( 𝑡 ) } = { 𝑡𝑇 ∣ 0 < ( ( ( 𝑌𝐹 ) ‘ 𝑖 ) ‘ 𝑡 ) } )
127 126 eqeq2d ( = ( ( 𝑌𝐹 ) ‘ 𝑖 ) → ( 𝑤 = { 𝑡𝑇 ∣ 0 < ( 𝑡 ) } ↔ 𝑤 = { 𝑡𝑇 ∣ 0 < ( ( ( 𝑌𝐹 ) ‘ 𝑖 ) ‘ 𝑡 ) } ) )
128 122 11 123 127 elrabf ( ( ( 𝑌𝐹 ) ‘ 𝑖 ) ∈ { 𝑄𝑤 = { 𝑡𝑇 ∣ 0 < ( 𝑡 ) } } ↔ ( ( ( 𝑌𝐹 ) ‘ 𝑖 ) ∈ 𝑄𝑤 = { 𝑡𝑇 ∣ 0 < ( ( ( 𝑌𝐹 ) ‘ 𝑖 ) ‘ 𝑡 ) } ) )
129 121 128 sylib ( ( ( 𝜑 ∧ ( 𝑡𝑤𝑤𝑋 ) ) ∧ ( ( 𝑌𝐹 ) ‘ 𝑖 ) ∈ ( 𝐺𝑤 ) ) → ( ( ( 𝑌𝐹 ) ‘ 𝑖 ) ∈ 𝑄𝑤 = { 𝑡𝑇 ∣ 0 < ( ( ( 𝑌𝐹 ) ‘ 𝑖 ) ‘ 𝑡 ) } ) )
130 129 simprd ( ( ( 𝜑 ∧ ( 𝑡𝑤𝑤𝑋 ) ) ∧ ( ( 𝑌𝐹 ) ‘ 𝑖 ) ∈ ( 𝐺𝑤 ) ) → 𝑤 = { 𝑡𝑇 ∣ 0 < ( ( ( 𝑌𝐹 ) ‘ 𝑖 ) ‘ 𝑡 ) } )
131 115 130 eleqtrd ( ( ( 𝜑 ∧ ( 𝑡𝑤𝑤𝑋 ) ) ∧ ( ( 𝑌𝐹 ) ‘ 𝑖 ) ∈ ( 𝐺𝑤 ) ) → 𝑡 ∈ { 𝑡𝑇 ∣ 0 < ( ( ( 𝑌𝐹 ) ‘ 𝑖 ) ‘ 𝑡 ) } )
132 rabid ( 𝑡 ∈ { 𝑡𝑇 ∣ 0 < ( ( ( 𝑌𝐹 ) ‘ 𝑖 ) ‘ 𝑡 ) } ↔ ( 𝑡𝑇 ∧ 0 < ( ( ( 𝑌𝐹 ) ‘ 𝑖 ) ‘ 𝑡 ) ) )
133 131 132 sylib ( ( ( 𝜑 ∧ ( 𝑡𝑤𝑤𝑋 ) ) ∧ ( ( 𝑌𝐹 ) ‘ 𝑖 ) ∈ ( 𝐺𝑤 ) ) → ( 𝑡𝑇 ∧ 0 < ( ( ( 𝑌𝐹 ) ‘ 𝑖 ) ‘ 𝑡 ) ) )
134 133 simprd ( ( ( 𝜑 ∧ ( 𝑡𝑤𝑤𝑋 ) ) ∧ ( ( 𝑌𝐹 ) ‘ 𝑖 ) ∈ ( 𝐺𝑤 ) ) → 0 < ( ( ( 𝑌𝐹 ) ‘ 𝑖 ) ‘ 𝑡 ) )
135 134 ex ( ( 𝜑 ∧ ( 𝑡𝑤𝑤𝑋 ) ) → ( ( ( 𝑌𝐹 ) ‘ 𝑖 ) ∈ ( 𝐺𝑤 ) → 0 < ( ( ( 𝑌𝐹 ) ‘ 𝑖 ) ‘ 𝑡 ) ) )
136 135 reximdv ( ( 𝜑 ∧ ( 𝑡𝑤𝑤𝑋 ) ) → ( ∃ 𝑖 ∈ ( 1 ... 𝑀 ) ( ( 𝑌𝐹 ) ‘ 𝑖 ) ∈ ( 𝐺𝑤 ) → ∃ 𝑖 ∈ ( 1 ... 𝑀 ) 0 < ( ( ( 𝑌𝐹 ) ‘ 𝑖 ) ‘ 𝑡 ) ) )
137 114 136 mpd ( ( 𝜑 ∧ ( 𝑡𝑤𝑤𝑋 ) ) → ∃ 𝑖 ∈ ( 1 ... 𝑀 ) 0 < ( ( ( 𝑌𝐹 ) ‘ 𝑖 ) ‘ 𝑡 ) )
138 137 ex ( 𝜑 → ( ( 𝑡𝑤𝑤𝑋 ) → ∃ 𝑖 ∈ ( 1 ... 𝑀 ) 0 < ( ( ( 𝑌𝐹 ) ‘ 𝑖 ) ‘ 𝑡 ) ) )
139 138 adantr ( ( 𝜑𝑡 ∈ ( 𝑇𝑈 ) ) → ( ( 𝑡𝑤𝑤𝑋 ) → ∃ 𝑖 ∈ ( 1 ... 𝑀 ) 0 < ( ( ( 𝑌𝐹 ) ‘ 𝑖 ) ‘ 𝑡 ) ) )
140 66 67 70 139 exlimimdd ( ( 𝜑𝑡 ∈ ( 𝑇𝑈 ) ) → ∃ 𝑖 ∈ ( 1 ... 𝑀 ) 0 < ( ( ( 𝑌𝐹 ) ‘ 𝑖 ) ‘ 𝑡 ) )
141 140 ex ( 𝜑 → ( 𝑡 ∈ ( 𝑇𝑈 ) → ∃ 𝑖 ∈ ( 1 ... 𝑀 ) 0 < ( ( ( 𝑌𝐹 ) ‘ 𝑖 ) ‘ 𝑡 ) ) )
142 9 141 ralrimi ( 𝜑 → ∀ 𝑡 ∈ ( 𝑇𝑈 ) ∃ 𝑖 ∈ ( 1 ... 𝑀 ) 0 < ( ( ( 𝑌𝐹 ) ‘ 𝑖 ) ‘ 𝑡 ) )
143 3 64 142 jca32 ( 𝜑 → ( 𝑀 ∈ ℕ ∧ ( ( 𝑌𝐹 ) : ( 1 ... 𝑀 ) ⟶ 𝑄 ∧ ∀ 𝑡 ∈ ( 𝑇𝑈 ) ∃ 𝑖 ∈ ( 1 ... 𝑀 ) 0 < ( ( ( 𝑌𝐹 ) ‘ 𝑖 ) ‘ 𝑡 ) ) ) )
144 feq1 ( 𝑞 = ( 𝑌𝐹 ) → ( 𝑞 : ( 1 ... 𝑀 ) ⟶ 𝑄 ↔ ( 𝑌𝐹 ) : ( 1 ... 𝑀 ) ⟶ 𝑄 ) )
145 fveq1 ( 𝑞 = ( 𝑌𝐹 ) → ( 𝑞𝑖 ) = ( ( 𝑌𝐹 ) ‘ 𝑖 ) )
146 145 fveq1d ( 𝑞 = ( 𝑌𝐹 ) → ( ( 𝑞𝑖 ) ‘ 𝑡 ) = ( ( ( 𝑌𝐹 ) ‘ 𝑖 ) ‘ 𝑡 ) )
147 146 breq2d ( 𝑞 = ( 𝑌𝐹 ) → ( 0 < ( ( 𝑞𝑖 ) ‘ 𝑡 ) ↔ 0 < ( ( ( 𝑌𝐹 ) ‘ 𝑖 ) ‘ 𝑡 ) ) )
148 147 rexbidv ( 𝑞 = ( 𝑌𝐹 ) → ( ∃ 𝑖 ∈ ( 1 ... 𝑀 ) 0 < ( ( 𝑞𝑖 ) ‘ 𝑡 ) ↔ ∃ 𝑖 ∈ ( 1 ... 𝑀 ) 0 < ( ( ( 𝑌𝐹 ) ‘ 𝑖 ) ‘ 𝑡 ) ) )
149 148 ralbidv ( 𝑞 = ( 𝑌𝐹 ) → ( ∀ 𝑡 ∈ ( 𝑇𝑈 ) ∃ 𝑖 ∈ ( 1 ... 𝑀 ) 0 < ( ( 𝑞𝑖 ) ‘ 𝑡 ) ↔ ∀ 𝑡 ∈ ( 𝑇𝑈 ) ∃ 𝑖 ∈ ( 1 ... 𝑀 ) 0 < ( ( ( 𝑌𝐹 ) ‘ 𝑖 ) ‘ 𝑡 ) ) )
150 144 149 anbi12d ( 𝑞 = ( 𝑌𝐹 ) → ( ( 𝑞 : ( 1 ... 𝑀 ) ⟶ 𝑄 ∧ ∀ 𝑡 ∈ ( 𝑇𝑈 ) ∃ 𝑖 ∈ ( 1 ... 𝑀 ) 0 < ( ( 𝑞𝑖 ) ‘ 𝑡 ) ) ↔ ( ( 𝑌𝐹 ) : ( 1 ... 𝑀 ) ⟶ 𝑄 ∧ ∀ 𝑡 ∈ ( 𝑇𝑈 ) ∃ 𝑖 ∈ ( 1 ... 𝑀 ) 0 < ( ( ( 𝑌𝐹 ) ‘ 𝑖 ) ‘ 𝑡 ) ) ) )
151 150 anbi2d ( 𝑞 = ( 𝑌𝐹 ) → ( ( 𝑀 ∈ ℕ ∧ ( 𝑞 : ( 1 ... 𝑀 ) ⟶ 𝑄 ∧ ∀ 𝑡 ∈ ( 𝑇𝑈 ) ∃ 𝑖 ∈ ( 1 ... 𝑀 ) 0 < ( ( 𝑞𝑖 ) ‘ 𝑡 ) ) ) ↔ ( 𝑀 ∈ ℕ ∧ ( ( 𝑌𝐹 ) : ( 1 ... 𝑀 ) ⟶ 𝑄 ∧ ∀ 𝑡 ∈ ( 𝑇𝑈 ) ∃ 𝑖 ∈ ( 1 ... 𝑀 ) 0 < ( ( ( 𝑌𝐹 ) ‘ 𝑖 ) ‘ 𝑡 ) ) ) ) )
152 151 spcegv ( ( 𝑌𝐹 ) ∈ V → ( ( 𝑀 ∈ ℕ ∧ ( ( 𝑌𝐹 ) : ( 1 ... 𝑀 ) ⟶ 𝑄 ∧ ∀ 𝑡 ∈ ( 𝑇𝑈 ) ∃ 𝑖 ∈ ( 1 ... 𝑀 ) 0 < ( ( ( 𝑌𝐹 ) ‘ 𝑖 ) ‘ 𝑡 ) ) ) → ∃ 𝑞 ( 𝑀 ∈ ℕ ∧ ( 𝑞 : ( 1 ... 𝑀 ) ⟶ 𝑄 ∧ ∀ 𝑡 ∈ ( 𝑇𝑈 ) ∃ 𝑖 ∈ ( 1 ... 𝑀 ) 0 < ( ( 𝑞𝑖 ) ‘ 𝑡 ) ) ) ) )
153 20 143 152 sylc ( 𝜑 → ∃ 𝑞 ( 𝑀 ∈ ℕ ∧ ( 𝑞 : ( 1 ... 𝑀 ) ⟶ 𝑄 ∧ ∀ 𝑡 ∈ ( 𝑇𝑈 ) ∃ 𝑖 ∈ ( 1 ... 𝑀 ) 0 < ( ( 𝑞𝑖 ) ‘ 𝑡 ) ) ) )