Step |
Hyp |
Ref |
Expression |
1 |
|
fcomptf.1 |
⊢ Ⅎ 𝑥 𝐵 |
2 |
|
nfcv |
⊢ Ⅎ 𝑥 𝐴 |
3 |
|
nfcv |
⊢ Ⅎ 𝑥 𝐷 |
4 |
|
nfcv |
⊢ Ⅎ 𝑥 𝐸 |
5 |
2 3 4
|
nff |
⊢ Ⅎ 𝑥 𝐴 : 𝐷 ⟶ 𝐸 |
6 |
|
nfcv |
⊢ Ⅎ 𝑥 𝐶 |
7 |
1 6 3
|
nff |
⊢ Ⅎ 𝑥 𝐵 : 𝐶 ⟶ 𝐷 |
8 |
5 7
|
nfan |
⊢ Ⅎ 𝑥 ( 𝐴 : 𝐷 ⟶ 𝐸 ∧ 𝐵 : 𝐶 ⟶ 𝐷 ) |
9 |
|
ffvelrn |
⊢ ( ( 𝐵 : 𝐶 ⟶ 𝐷 ∧ 𝑥 ∈ 𝐶 ) → ( 𝐵 ‘ 𝑥 ) ∈ 𝐷 ) |
10 |
9
|
adantll |
⊢ ( ( ( 𝐴 : 𝐷 ⟶ 𝐸 ∧ 𝐵 : 𝐶 ⟶ 𝐷 ) ∧ 𝑥 ∈ 𝐶 ) → ( 𝐵 ‘ 𝑥 ) ∈ 𝐷 ) |
11 |
10
|
ex |
⊢ ( ( 𝐴 : 𝐷 ⟶ 𝐸 ∧ 𝐵 : 𝐶 ⟶ 𝐷 ) → ( 𝑥 ∈ 𝐶 → ( 𝐵 ‘ 𝑥 ) ∈ 𝐷 ) ) |
12 |
8 11
|
ralrimi |
⊢ ( ( 𝐴 : 𝐷 ⟶ 𝐸 ∧ 𝐵 : 𝐶 ⟶ 𝐷 ) → ∀ 𝑥 ∈ 𝐶 ( 𝐵 ‘ 𝑥 ) ∈ 𝐷 ) |
13 |
|
ffn |
⊢ ( 𝐵 : 𝐶 ⟶ 𝐷 → 𝐵 Fn 𝐶 ) |
14 |
13
|
adantl |
⊢ ( ( 𝐴 : 𝐷 ⟶ 𝐸 ∧ 𝐵 : 𝐶 ⟶ 𝐷 ) → 𝐵 Fn 𝐶 ) |
15 |
1
|
dffn5f |
⊢ ( 𝐵 Fn 𝐶 ↔ 𝐵 = ( 𝑥 ∈ 𝐶 ↦ ( 𝐵 ‘ 𝑥 ) ) ) |
16 |
14 15
|
sylib |
⊢ ( ( 𝐴 : 𝐷 ⟶ 𝐸 ∧ 𝐵 : 𝐶 ⟶ 𝐷 ) → 𝐵 = ( 𝑥 ∈ 𝐶 ↦ ( 𝐵 ‘ 𝑥 ) ) ) |
17 |
|
ffn |
⊢ ( 𝐴 : 𝐷 ⟶ 𝐸 → 𝐴 Fn 𝐷 ) |
18 |
17
|
adantr |
⊢ ( ( 𝐴 : 𝐷 ⟶ 𝐸 ∧ 𝐵 : 𝐶 ⟶ 𝐷 ) → 𝐴 Fn 𝐷 ) |
19 |
|
dffn5 |
⊢ ( 𝐴 Fn 𝐷 ↔ 𝐴 = ( 𝑦 ∈ 𝐷 ↦ ( 𝐴 ‘ 𝑦 ) ) ) |
20 |
18 19
|
sylib |
⊢ ( ( 𝐴 : 𝐷 ⟶ 𝐸 ∧ 𝐵 : 𝐶 ⟶ 𝐷 ) → 𝐴 = ( 𝑦 ∈ 𝐷 ↦ ( 𝐴 ‘ 𝑦 ) ) ) |
21 |
|
fveq2 |
⊢ ( 𝑦 = ( 𝐵 ‘ 𝑥 ) → ( 𝐴 ‘ 𝑦 ) = ( 𝐴 ‘ ( 𝐵 ‘ 𝑥 ) ) ) |
22 |
12 16 20 21
|
fmptcof |
⊢ ( ( 𝐴 : 𝐷 ⟶ 𝐸 ∧ 𝐵 : 𝐶 ⟶ 𝐷 ) → ( 𝐴 ∘ 𝐵 ) = ( 𝑥 ∈ 𝐶 ↦ ( 𝐴 ‘ ( 𝐵 ‘ 𝑥 ) ) ) ) |