Step |
Hyp |
Ref |
Expression |
1 |
|
climeldmeq.z |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
2 |
|
climeldmeq.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝑉 ) |
3 |
|
climeldmeq.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝑊 ) |
4 |
|
climeldmeq.m |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
5 |
|
climeldmeq.e |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) = ( 𝐺 ‘ 𝑘 ) ) |
6 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐺 ∈ 𝑊 ) |
7 |
|
fvexd |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ dom ⇝ ) → ( ⇝ ‘ 𝐹 ) ∈ V ) |
8 |
|
climdm |
⊢ ( 𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ ( ⇝ ‘ 𝐹 ) ) |
9 |
8
|
a1i |
⊢ ( 𝜑 → ( 𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ ( ⇝ ‘ 𝐹 ) ) ) |
10 |
9
|
biimpa |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ⇝ ( ⇝ ‘ 𝐹 ) ) |
11 |
1 2 3 4 5
|
climeq |
⊢ ( 𝜑 → ( 𝐹 ⇝ ( ⇝ ‘ 𝐹 ) ↔ 𝐺 ⇝ ( ⇝ ‘ 𝐹 ) ) ) |
12 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ dom ⇝ ) → ( 𝐹 ⇝ ( ⇝ ‘ 𝐹 ) ↔ 𝐺 ⇝ ( ⇝ ‘ 𝐹 ) ) ) |
13 |
10 12
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐺 ⇝ ( ⇝ ‘ 𝐹 ) ) |
14 |
|
breldmg |
⊢ ( ( 𝐺 ∈ 𝑊 ∧ ( ⇝ ‘ 𝐹 ) ∈ V ∧ 𝐺 ⇝ ( ⇝ ‘ 𝐹 ) ) → 𝐺 ∈ dom ⇝ ) |
15 |
6 7 13 14
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐺 ∈ dom ⇝ ) |
16 |
15
|
ex |
⊢ ( 𝜑 → ( 𝐹 ∈ dom ⇝ → 𝐺 ∈ dom ⇝ ) ) |
17 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐺 ∈ dom ⇝ ) → 𝐹 ∈ 𝑉 ) |
18 |
|
fvexd |
⊢ ( ( 𝜑 ∧ 𝐺 ∈ dom ⇝ ) → ( ⇝ ‘ 𝐺 ) ∈ V ) |
19 |
|
climdm |
⊢ ( 𝐺 ∈ dom ⇝ ↔ 𝐺 ⇝ ( ⇝ ‘ 𝐺 ) ) |
20 |
19
|
biimpi |
⊢ ( 𝐺 ∈ dom ⇝ → 𝐺 ⇝ ( ⇝ ‘ 𝐺 ) ) |
21 |
20
|
adantl |
⊢ ( ( 𝜑 ∧ 𝐺 ∈ dom ⇝ ) → 𝐺 ⇝ ( ⇝ ‘ 𝐺 ) ) |
22 |
5
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐺 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑘 ) ) |
23 |
1 3 2 4 22
|
climeq |
⊢ ( 𝜑 → ( 𝐺 ⇝ ( ⇝ ‘ 𝐺 ) ↔ 𝐹 ⇝ ( ⇝ ‘ 𝐺 ) ) ) |
24 |
23
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐺 ∈ dom ⇝ ) → ( 𝐺 ⇝ ( ⇝ ‘ 𝐺 ) ↔ 𝐹 ⇝ ( ⇝ ‘ 𝐺 ) ) ) |
25 |
21 24
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝐺 ∈ dom ⇝ ) → 𝐹 ⇝ ( ⇝ ‘ 𝐺 ) ) |
26 |
|
breldmg |
⊢ ( ( 𝐹 ∈ 𝑉 ∧ ( ⇝ ‘ 𝐺 ) ∈ V ∧ 𝐹 ⇝ ( ⇝ ‘ 𝐺 ) ) → 𝐹 ∈ dom ⇝ ) |
27 |
17 18 25 26
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝐺 ∈ dom ⇝ ) → 𝐹 ∈ dom ⇝ ) |
28 |
27
|
ex |
⊢ ( 𝜑 → ( 𝐺 ∈ dom ⇝ → 𝐹 ∈ dom ⇝ ) ) |
29 |
16 28
|
impbid |
⊢ ( 𝜑 → ( 𝐹 ∈ dom ⇝ ↔ 𝐺 ∈ dom ⇝ ) ) |