Step |
Hyp |
Ref |
Expression |
1 |
|
fnlimfvre.p |
⊢ Ⅎ 𝑚 𝜑 |
2 |
|
fnlimfvre.m |
⊢ Ⅎ 𝑚 𝐹 |
3 |
|
fnlimfvre.n |
⊢ Ⅎ 𝑥 𝐹 |
4 |
|
fnlimfvre.z |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
5 |
|
fnlimfvre.f |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑚 ) : dom ( 𝐹 ‘ 𝑚 ) ⟶ ℝ ) |
6 |
|
fnlimfvre.d |
⊢ 𝐷 = { 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∣ ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) ∈ dom ⇝ } |
7 |
|
fnlimfvre.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐷 ) |
8 |
|
nfcv |
⊢ Ⅎ 𝑥 𝑍 |
9 |
|
nfcv |
⊢ Ⅎ 𝑥 ( ℤ≥ ‘ 𝑛 ) |
10 |
|
nfcv |
⊢ Ⅎ 𝑥 𝑚 |
11 |
3 10
|
nffv |
⊢ Ⅎ 𝑥 ( 𝐹 ‘ 𝑚 ) |
12 |
11
|
nfdm |
⊢ Ⅎ 𝑥 dom ( 𝐹 ‘ 𝑚 ) |
13 |
9 12
|
nfiin |
⊢ Ⅎ 𝑥 ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) |
14 |
8 13
|
nfiun |
⊢ Ⅎ 𝑥 ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) |
15 |
14
|
ssrab2f |
⊢ { 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∣ ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) ∈ dom ⇝ } ⊆ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) |
16 |
6 15
|
eqsstri |
⊢ 𝐷 ⊆ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) |
17 |
16
|
sseli |
⊢ ( 𝑋 ∈ 𝐷 → 𝑋 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ) |
18 |
|
eliun |
⊢ ( 𝑋 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ↔ ∃ 𝑛 ∈ 𝑍 𝑋 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ) |
19 |
17 18
|
sylib |
⊢ ( 𝑋 ∈ 𝐷 → ∃ 𝑛 ∈ 𝑍 𝑋 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ) |
20 |
7 19
|
syl |
⊢ ( 𝜑 → ∃ 𝑛 ∈ 𝑍 𝑋 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ) |
21 |
|
nfv |
⊢ Ⅎ 𝑛 𝜑 |
22 |
|
nfv |
⊢ Ⅎ 𝑛 ( ⇝ ‘ ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ) ∈ ℝ |
23 |
|
nfv |
⊢ Ⅎ 𝑚 𝑛 ∈ 𝑍 |
24 |
|
nfcv |
⊢ Ⅎ 𝑚 𝑋 |
25 |
|
nfii1 |
⊢ Ⅎ 𝑚 ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) |
26 |
24 25
|
nfel |
⊢ Ⅎ 𝑚 𝑋 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) |
27 |
1 23 26
|
nf3an |
⊢ Ⅎ 𝑚 ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ) |
28 |
|
uzssz |
⊢ ( ℤ≥ ‘ 𝑀 ) ⊆ ℤ |
29 |
4
|
eleq2i |
⊢ ( 𝑛 ∈ 𝑍 ↔ 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
30 |
29
|
biimpi |
⊢ ( 𝑛 ∈ 𝑍 → 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
31 |
28 30
|
sseldi |
⊢ ( 𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ ) |
32 |
31
|
3ad2ant2 |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ) → 𝑛 ∈ ℤ ) |
33 |
|
eqid |
⊢ ( ℤ≥ ‘ 𝑛 ) = ( ℤ≥ ‘ 𝑛 ) |
34 |
4
|
fvexi |
⊢ 𝑍 ∈ V |
35 |
34
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ) → 𝑍 ∈ V ) |
36 |
4
|
uztrn2 |
⊢ ( ( 𝑛 ∈ 𝑍 ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝑗 ∈ 𝑍 ) |
37 |
36
|
ssd |
⊢ ( 𝑛 ∈ 𝑍 → ( ℤ≥ ‘ 𝑛 ) ⊆ 𝑍 ) |
38 |
37
|
3ad2ant2 |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ) → ( ℤ≥ ‘ 𝑛 ) ⊆ 𝑍 ) |
39 |
|
fvexd |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ) ∧ 𝑚 ∈ 𝑍 ) → ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ∈ V ) |
40 |
|
fvexd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ) → ( ℤ≥ ‘ 𝑛 ) ∈ V ) |
41 |
|
ssidd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ) → ( ℤ≥ ‘ 𝑛 ) ⊆ ( ℤ≥ ‘ 𝑛 ) ) |
42 |
|
fvexd |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ∈ V ) |
43 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) = ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) |
44 |
27 32 33 35 38 39 40 41 42 43
|
climfveqmpt |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ) → ( ⇝ ‘ ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ) = ( ⇝ ‘ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ) ) |
45 |
6
|
eleq2i |
⊢ ( 𝑋 ∈ 𝐷 ↔ 𝑋 ∈ { 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∣ ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) ∈ dom ⇝ } ) |
46 |
45
|
biimpi |
⊢ ( 𝑋 ∈ 𝐷 → 𝑋 ∈ { 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∣ ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) ∈ dom ⇝ } ) |
47 |
|
nfcv |
⊢ Ⅎ 𝑥 𝑋 |
48 |
11 47
|
nffv |
⊢ Ⅎ 𝑥 ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) |
49 |
8 48
|
nfmpt |
⊢ Ⅎ 𝑥 ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) |
50 |
|
nfcv |
⊢ Ⅎ 𝑥 dom ⇝ |
51 |
49 50
|
nfel |
⊢ Ⅎ 𝑥 ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ∈ dom ⇝ |
52 |
|
fveq2 |
⊢ ( 𝑥 = 𝑋 → ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) = ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) |
53 |
52
|
mpteq2dv |
⊢ ( 𝑥 = 𝑋 → ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) = ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ) |
54 |
53
|
eleq1d |
⊢ ( 𝑥 = 𝑋 → ( ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) ∈ dom ⇝ ↔ ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ∈ dom ⇝ ) ) |
55 |
47 14 51 54
|
elrabf |
⊢ ( 𝑋 ∈ { 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∣ ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) ∈ dom ⇝ } ↔ ( 𝑋 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∧ ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ∈ dom ⇝ ) ) |
56 |
55
|
biimpi |
⊢ ( 𝑋 ∈ { 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∣ ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) ∈ dom ⇝ } → ( 𝑋 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∧ ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ∈ dom ⇝ ) ) |
57 |
56
|
simprd |
⊢ ( 𝑋 ∈ { 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∣ ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) ∈ dom ⇝ } → ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ∈ dom ⇝ ) |
58 |
46 57
|
syl |
⊢ ( 𝑋 ∈ 𝐷 → ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ∈ dom ⇝ ) |
59 |
58
|
adantr |
⊢ ( ( 𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ∈ dom ⇝ ) |
60 |
|
nfmpt1 |
⊢ Ⅎ 𝑚 ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) |
61 |
|
nfcv |
⊢ Ⅎ 𝑚 dom ⇝ |
62 |
60 61
|
nfel |
⊢ Ⅎ 𝑚 ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) ∈ dom ⇝ |
63 |
|
nfv |
⊢ Ⅎ 𝑚 𝑗 ∈ 𝑍 |
64 |
63
|
nfci |
⊢ Ⅎ 𝑚 𝑍 |
65 |
64 25
|
nfiun |
⊢ Ⅎ 𝑚 ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) |
66 |
62 65
|
nfrabw |
⊢ Ⅎ 𝑚 { 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∣ ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑥 ) ) ∈ dom ⇝ } |
67 |
6 66
|
nfcxfr |
⊢ Ⅎ 𝑚 𝐷 |
68 |
24 67
|
nfel |
⊢ Ⅎ 𝑚 𝑋 ∈ 𝐷 |
69 |
68 23
|
nfan |
⊢ Ⅎ 𝑚 ( 𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍 ) |
70 |
31
|
adantl |
⊢ ( ( 𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍 ) → 𝑛 ∈ ℤ ) |
71 |
34
|
a1i |
⊢ ( ( 𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍 ) → 𝑍 ∈ V ) |
72 |
37
|
adantl |
⊢ ( ( 𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍 ) → ( ℤ≥ ‘ 𝑛 ) ⊆ 𝑍 ) |
73 |
|
fvexd |
⊢ ( ( ( 𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑚 ∈ 𝑍 ) → ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ∈ V ) |
74 |
|
fvexd |
⊢ ( ( 𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍 ) → ( ℤ≥ ‘ 𝑛 ) ∈ V ) |
75 |
|
ssidd |
⊢ ( ( 𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍 ) → ( ℤ≥ ‘ 𝑛 ) ⊆ ( ℤ≥ ‘ 𝑛 ) ) |
76 |
|
fvexd |
⊢ ( ( ( 𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ∈ V ) |
77 |
|
eqidd |
⊢ ( ( ( 𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) = ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) |
78 |
69 70 33 71 72 73 74 75 76 77
|
climeldmeqmpt |
⊢ ( ( 𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍 ) → ( ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ∈ dom ⇝ ↔ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ∈ dom ⇝ ) ) |
79 |
59 78
|
mpbid |
⊢ ( ( 𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ∈ dom ⇝ ) |
80 |
|
climdm |
⊢ ( ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ∈ dom ⇝ ↔ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ⇝ ( ⇝ ‘ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ) ) |
81 |
79 80
|
sylib |
⊢ ( ( 𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ⇝ ( ⇝ ‘ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ) ) |
82 |
7 81
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ⇝ ( ⇝ ‘ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ) ) |
83 |
82
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ) → ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ⇝ ( ⇝ ‘ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ) ) |
84 |
|
simpl1 |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ) ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝜑 ) |
85 |
|
simpl2 |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ) ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝑛 ∈ 𝑍 ) |
86 |
|
nfcv |
⊢ Ⅎ 𝑗 dom ( 𝐹 ‘ 𝑚 ) |
87 |
|
nfcv |
⊢ Ⅎ 𝑚 𝑗 |
88 |
2 87
|
nffv |
⊢ Ⅎ 𝑚 ( 𝐹 ‘ 𝑗 ) |
89 |
88
|
nfdm |
⊢ Ⅎ 𝑚 dom ( 𝐹 ‘ 𝑗 ) |
90 |
|
fveq2 |
⊢ ( 𝑚 = 𝑗 → ( 𝐹 ‘ 𝑚 ) = ( 𝐹 ‘ 𝑗 ) ) |
91 |
90
|
dmeqd |
⊢ ( 𝑚 = 𝑗 → dom ( 𝐹 ‘ 𝑚 ) = dom ( 𝐹 ‘ 𝑗 ) ) |
92 |
86 89 91
|
cbviin |
⊢ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) = ∩ 𝑗 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑗 ) |
93 |
92
|
eleq2i |
⊢ ( 𝑋 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ↔ 𝑋 ∈ ∩ 𝑗 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑗 ) ) |
94 |
93
|
biimpi |
⊢ ( 𝑋 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) → 𝑋 ∈ ∩ 𝑗 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑗 ) ) |
95 |
94
|
adantr |
⊢ ( ( 𝑋 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝑋 ∈ ∩ 𝑗 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑗 ) ) |
96 |
|
simpr |
⊢ ( ( 𝑋 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝑗 ∈ ( ℤ≥ ‘ 𝑛 ) ) |
97 |
|
eliinid |
⊢ ( ( 𝑋 ∈ ∩ 𝑗 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑗 ) ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝑋 ∈ dom ( 𝐹 ‘ 𝑗 ) ) |
98 |
95 96 97
|
syl2anc |
⊢ ( ( 𝑋 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝑋 ∈ dom ( 𝐹 ‘ 𝑗 ) ) |
99 |
98
|
3ad2antl3 |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ) ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝑋 ∈ dom ( 𝐹 ‘ 𝑗 ) ) |
100 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ) ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝑗 ∈ ( ℤ≥ ‘ 𝑛 ) ) |
101 |
|
id |
⊢ ( 𝑗 ∈ ( ℤ≥ ‘ 𝑛 ) → 𝑗 ∈ ( ℤ≥ ‘ 𝑛 ) ) |
102 |
|
fvexd |
⊢ ( 𝑗 ∈ ( ℤ≥ ‘ 𝑛 ) → ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑋 ) ∈ V ) |
103 |
88 24
|
nffv |
⊢ Ⅎ 𝑚 ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑋 ) |
104 |
90
|
fveq1d |
⊢ ( 𝑚 = 𝑗 → ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) = ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑋 ) ) |
105 |
|
eqid |
⊢ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) = ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) |
106 |
87 103 104 105
|
fvmptf |
⊢ ( ( 𝑗 ∈ ( ℤ≥ ‘ 𝑛 ) ∧ ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑋 ) ∈ V ) → ( ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ‘ 𝑗 ) = ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑋 ) ) |
107 |
101 102 106
|
syl2anc |
⊢ ( 𝑗 ∈ ( ℤ≥ ‘ 𝑛 ) → ( ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ‘ 𝑗 ) = ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑋 ) ) |
108 |
107
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ dom ( 𝐹 ‘ 𝑗 ) ) ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ‘ 𝑗 ) = ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑋 ) ) |
109 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝜑 ) |
110 |
36
|
adantll |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝑗 ∈ 𝑍 ) |
111 |
1 63
|
nfan |
⊢ Ⅎ 𝑚 ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) |
112 |
|
nfcv |
⊢ Ⅎ 𝑚 ℝ |
113 |
88 89 112
|
nff |
⊢ Ⅎ 𝑚 ( 𝐹 ‘ 𝑗 ) : dom ( 𝐹 ‘ 𝑗 ) ⟶ ℝ |
114 |
111 113
|
nfim |
⊢ Ⅎ 𝑚 ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑗 ) : dom ( 𝐹 ‘ 𝑗 ) ⟶ ℝ ) |
115 |
|
eleq1w |
⊢ ( 𝑚 = 𝑗 → ( 𝑚 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍 ) ) |
116 |
115
|
anbi2d |
⊢ ( 𝑚 = 𝑗 → ( ( 𝜑 ∧ 𝑚 ∈ 𝑍 ) ↔ ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ) ) |
117 |
90 91
|
feq12d |
⊢ ( 𝑚 = 𝑗 → ( ( 𝐹 ‘ 𝑚 ) : dom ( 𝐹 ‘ 𝑚 ) ⟶ ℝ ↔ ( 𝐹 ‘ 𝑗 ) : dom ( 𝐹 ‘ 𝑗 ) ⟶ ℝ ) ) |
118 |
116 117
|
imbi12d |
⊢ ( 𝑚 = 𝑗 → ( ( ( 𝜑 ∧ 𝑚 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑚 ) : dom ( 𝐹 ‘ 𝑚 ) ⟶ ℝ ) ↔ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑗 ) : dom ( 𝐹 ‘ 𝑗 ) ⟶ ℝ ) ) ) |
119 |
114 118 5
|
chvarfv |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑗 ) : dom ( 𝐹 ‘ 𝑗 ) ⟶ ℝ ) |
120 |
109 110 119
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( 𝐹 ‘ 𝑗 ) : dom ( 𝐹 ‘ 𝑗 ) ⟶ ℝ ) |
121 |
120
|
3adantl3 |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ dom ( 𝐹 ‘ 𝑗 ) ) ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( 𝐹 ‘ 𝑗 ) : dom ( 𝐹 ‘ 𝑗 ) ⟶ ℝ ) |
122 |
|
simpl3 |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ dom ( 𝐹 ‘ 𝑗 ) ) ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑛 ) ) → 𝑋 ∈ dom ( 𝐹 ‘ 𝑗 ) ) |
123 |
121 122
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ dom ( 𝐹 ‘ 𝑗 ) ) ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑋 ) ∈ ℝ ) |
124 |
108 123
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ dom ( 𝐹 ‘ 𝑗 ) ) ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ‘ 𝑗 ) ∈ ℝ ) |
125 |
84 85 99 100 124
|
syl31anc |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ) ∧ 𝑗 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ‘ 𝑗 ) ∈ ℝ ) |
126 |
33 32 83 125
|
climrecl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ) → ( ⇝ ‘ ( 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ) ∈ ℝ ) |
127 |
44 126
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) ) → ( ⇝ ‘ ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ) ∈ ℝ ) |
128 |
127
|
3exp |
⊢ ( 𝜑 → ( 𝑛 ∈ 𝑍 → ( 𝑋 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) → ( ⇝ ‘ ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ) ∈ ℝ ) ) ) |
129 |
21 22 128
|
rexlimd |
⊢ ( 𝜑 → ( ∃ 𝑛 ∈ 𝑍 𝑋 ∈ ∩ 𝑚 ∈ ( ℤ≥ ‘ 𝑛 ) dom ( 𝐹 ‘ 𝑚 ) → ( ⇝ ‘ ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ) ∈ ℝ ) ) |
130 |
20 129
|
mpd |
⊢ ( 𝜑 → ( ⇝ ‘ ( 𝑚 ∈ 𝑍 ↦ ( ( 𝐹 ‘ 𝑚 ) ‘ 𝑋 ) ) ) ∈ ℝ ) |