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