Step |
Hyp |
Ref |
Expression |
1 |
|
climeldmeqf.p |
⊢ Ⅎ 𝑘 𝜑 |
2 |
|
climeldmeqf.n |
⊢ Ⅎ 𝑘 𝐹 |
3 |
|
climeldmeqf.o |
⊢ Ⅎ 𝑘 𝐺 |
4 |
|
climeldmeqf.z |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
5 |
|
climeldmeqf.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝑉 ) |
6 |
|
climeldmeqf.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝑊 ) |
7 |
|
climeldmeqf.m |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
8 |
|
climeldmeqf.e |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) = ( 𝐺 ‘ 𝑘 ) ) |
9 |
|
nfv |
⊢ Ⅎ 𝑘 𝑗 ∈ 𝑍 |
10 |
1 9
|
nfan |
⊢ Ⅎ 𝑘 ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) |
11 |
|
nfcv |
⊢ Ⅎ 𝑘 𝑗 |
12 |
2 11
|
nffv |
⊢ Ⅎ 𝑘 ( 𝐹 ‘ 𝑗 ) |
13 |
3 11
|
nffv |
⊢ Ⅎ 𝑘 ( 𝐺 ‘ 𝑗 ) |
14 |
12 13
|
nfeq |
⊢ Ⅎ 𝑘 ( 𝐹 ‘ 𝑗 ) = ( 𝐺 ‘ 𝑗 ) |
15 |
10 14
|
nfim |
⊢ Ⅎ 𝑘 ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑗 ) = ( 𝐺 ‘ 𝑗 ) ) |
16 |
|
eleq1w |
⊢ ( 𝑘 = 𝑗 → ( 𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍 ) ) |
17 |
16
|
anbi2d |
⊢ ( 𝑘 = 𝑗 → ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ↔ ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ) ) |
18 |
|
fveq2 |
⊢ ( 𝑘 = 𝑗 → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑗 ) ) |
19 |
|
fveq2 |
⊢ ( 𝑘 = 𝑗 → ( 𝐺 ‘ 𝑘 ) = ( 𝐺 ‘ 𝑗 ) ) |
20 |
18 19
|
eqeq12d |
⊢ ( 𝑘 = 𝑗 → ( ( 𝐹 ‘ 𝑘 ) = ( 𝐺 ‘ 𝑘 ) ↔ ( 𝐹 ‘ 𝑗 ) = ( 𝐺 ‘ 𝑗 ) ) ) |
21 |
17 20
|
imbi12d |
⊢ ( 𝑘 = 𝑗 → ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) = ( 𝐺 ‘ 𝑘 ) ) ↔ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑗 ) = ( 𝐺 ‘ 𝑗 ) ) ) ) |
22 |
15 21 8
|
chvarfv |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑗 ) = ( 𝐺 ‘ 𝑗 ) ) |
23 |
4 5 6 7 22
|
climeldmeq |
⊢ ( 𝜑 → ( 𝐹 ∈ dom ⇝ ↔ 𝐺 ∈ dom ⇝ ) ) |