Step |
Hyp |
Ref |
Expression |
1 |
|
climfveqf.p |
⊢ Ⅎ 𝑘 𝜑 |
2 |
|
climfveqf.n |
⊢ Ⅎ 𝑘 𝐹 |
3 |
|
climfveqf.o |
⊢ Ⅎ 𝑘 𝐺 |
4 |
|
climfveqf.z |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
5 |
|
climfveqf.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝑉 ) |
6 |
|
climfveqf.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝑊 ) |
7 |
|
climfveqf.m |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
8 |
|
climfveqf.e |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) = ( 𝐺 ‘ 𝑘 ) ) |
9 |
|
climdm |
⊢ ( 𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ ( ⇝ ‘ 𝐹 ) ) |
10 |
9
|
biimpi |
⊢ ( 𝐹 ∈ dom ⇝ → 𝐹 ⇝ ( ⇝ ‘ 𝐹 ) ) |
11 |
10
|
adantl |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ⇝ ( ⇝ ‘ 𝐹 ) ) |
12 |
11 9
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ∈ dom ⇝ ) |
13 |
|
nfcv |
⊢ Ⅎ 𝑘 𝑗 |
14 |
13
|
nfel1 |
⊢ Ⅎ 𝑘 𝑗 ∈ 𝑍 |
15 |
1 14
|
nfan |
⊢ Ⅎ 𝑘 ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) |
16 |
2 13
|
nffv |
⊢ Ⅎ 𝑘 ( 𝐹 ‘ 𝑗 ) |
17 |
3 13
|
nffv |
⊢ Ⅎ 𝑘 ( 𝐺 ‘ 𝑗 ) |
18 |
16 17
|
nfeq |
⊢ Ⅎ 𝑘 ( 𝐹 ‘ 𝑗 ) = ( 𝐺 ‘ 𝑗 ) |
19 |
15 18
|
nfim |
⊢ Ⅎ 𝑘 ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑗 ) = ( 𝐺 ‘ 𝑗 ) ) |
20 |
|
eleq1w |
⊢ ( 𝑘 = 𝑗 → ( 𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍 ) ) |
21 |
20
|
anbi2d |
⊢ ( 𝑘 = 𝑗 → ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ↔ ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ) ) |
22 |
|
fveq2 |
⊢ ( 𝑘 = 𝑗 → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑗 ) ) |
23 |
|
fveq2 |
⊢ ( 𝑘 = 𝑗 → ( 𝐺 ‘ 𝑘 ) = ( 𝐺 ‘ 𝑗 ) ) |
24 |
22 23
|
eqeq12d |
⊢ ( 𝑘 = 𝑗 → ( ( 𝐹 ‘ 𝑘 ) = ( 𝐺 ‘ 𝑘 ) ↔ ( 𝐹 ‘ 𝑗 ) = ( 𝐺 ‘ 𝑗 ) ) ) |
25 |
21 24
|
imbi12d |
⊢ ( 𝑘 = 𝑗 → ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) = ( 𝐺 ‘ 𝑘 ) ) ↔ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑗 ) = ( 𝐺 ‘ 𝑗 ) ) ) ) |
26 |
19 25 8
|
chvarfv |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑗 ) = ( 𝐺 ‘ 𝑗 ) ) |
27 |
4 5 6 7 26
|
climeldmeq |
⊢ ( 𝜑 → ( 𝐹 ∈ dom ⇝ ↔ 𝐺 ∈ dom ⇝ ) ) |
28 |
27
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ dom ⇝ ) → ( 𝐹 ∈ dom ⇝ ↔ 𝐺 ∈ dom ⇝ ) ) |
29 |
12 28
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐺 ∈ dom ⇝ ) |
30 |
|
climdm |
⊢ ( 𝐺 ∈ dom ⇝ ↔ 𝐺 ⇝ ( ⇝ ‘ 𝐺 ) ) |
31 |
29 30
|
sylib |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐺 ⇝ ( ⇝ ‘ 𝐺 ) ) |
32 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐺 ∈ 𝑊 ) |
33 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ∈ 𝑉 ) |
34 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝑀 ∈ ℤ ) |
35 |
26
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( 𝐺 ‘ 𝑗 ) = ( 𝐹 ‘ 𝑗 ) ) |
36 |
35
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝐹 ∈ dom ⇝ ) ∧ 𝑗 ∈ 𝑍 ) → ( 𝐺 ‘ 𝑗 ) = ( 𝐹 ‘ 𝑗 ) ) |
37 |
4 32 33 34 36
|
climeq |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ dom ⇝ ) → ( 𝐺 ⇝ ( ⇝ ‘ 𝐺 ) ↔ 𝐹 ⇝ ( ⇝ ‘ 𝐺 ) ) ) |
38 |
31 37
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ⇝ ( ⇝ ‘ 𝐺 ) ) |
39 |
|
climuni |
⊢ ( ( 𝐹 ⇝ ( ⇝ ‘ 𝐹 ) ∧ 𝐹 ⇝ ( ⇝ ‘ 𝐺 ) ) → ( ⇝ ‘ 𝐹 ) = ( ⇝ ‘ 𝐺 ) ) |
40 |
11 38 39
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ dom ⇝ ) → ( ⇝ ‘ 𝐹 ) = ( ⇝ ‘ 𝐺 ) ) |
41 |
|
ndmfv |
⊢ ( ¬ 𝐹 ∈ dom ⇝ → ( ⇝ ‘ 𝐹 ) = ∅ ) |
42 |
41
|
adantl |
⊢ ( ( 𝜑 ∧ ¬ 𝐹 ∈ dom ⇝ ) → ( ⇝ ‘ 𝐹 ) = ∅ ) |
43 |
|
simpr |
⊢ ( ( 𝜑 ∧ ¬ 𝐹 ∈ dom ⇝ ) → ¬ 𝐹 ∈ dom ⇝ ) |
44 |
27
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐹 ∈ dom ⇝ ) → ( 𝐹 ∈ dom ⇝ ↔ 𝐺 ∈ dom ⇝ ) ) |
45 |
43 44
|
mtbid |
⊢ ( ( 𝜑 ∧ ¬ 𝐹 ∈ dom ⇝ ) → ¬ 𝐺 ∈ dom ⇝ ) |
46 |
|
ndmfv |
⊢ ( ¬ 𝐺 ∈ dom ⇝ → ( ⇝ ‘ 𝐺 ) = ∅ ) |
47 |
45 46
|
syl |
⊢ ( ( 𝜑 ∧ ¬ 𝐹 ∈ dom ⇝ ) → ( ⇝ ‘ 𝐺 ) = ∅ ) |
48 |
42 47
|
eqtr4d |
⊢ ( ( 𝜑 ∧ ¬ 𝐹 ∈ dom ⇝ ) → ( ⇝ ‘ 𝐹 ) = ( ⇝ ‘ 𝐺 ) ) |
49 |
40 48
|
pm2.61dan |
⊢ ( 𝜑 → ( ⇝ ‘ 𝐹 ) = ( ⇝ ‘ 𝐺 ) ) |