Step |
Hyp |
Ref |
Expression |
1 |
|
funimaeq.x |
⊢ Ⅎ 𝑥 𝜑 |
2 |
|
funimaeq.f |
⊢ ( 𝜑 → Fun 𝐹 ) |
3 |
|
funimaeq.g |
⊢ ( 𝜑 → Fun 𝐺 ) |
4 |
|
funimaeq.a |
⊢ ( 𝜑 → 𝐴 ⊆ dom 𝐹 ) |
5 |
|
funimaeq.d |
⊢ ( 𝜑 → 𝐴 ⊆ dom 𝐺 ) |
6 |
|
funimaeq.e |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑥 ) ) |
7 |
3
|
funfnd |
⊢ ( 𝜑 → 𝐺 Fn dom 𝐺 ) |
8 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐺 Fn dom 𝐺 ) |
9 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐴 ⊆ dom 𝐺 ) |
10 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ 𝐴 ) |
11 |
|
fnfvima |
⊢ ( ( 𝐺 Fn dom 𝐺 ∧ 𝐴 ⊆ dom 𝐺 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐺 ‘ 𝑥 ) ∈ ( 𝐺 “ 𝐴 ) ) |
12 |
8 9 10 11
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐺 ‘ 𝑥 ) ∈ ( 𝐺 “ 𝐴 ) ) |
13 |
6 12
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑥 ) ∈ ( 𝐺 “ 𝐴 ) ) |
14 |
1 2 13
|
funimassd |
⊢ ( 𝜑 → ( 𝐹 “ 𝐴 ) ⊆ ( 𝐺 “ 𝐴 ) ) |
15 |
2
|
funfnd |
⊢ ( 𝜑 → 𝐹 Fn dom 𝐹 ) |
16 |
15
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐹 Fn dom 𝐹 ) |
17 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐴 ⊆ dom 𝐹 ) |
18 |
|
fnfvima |
⊢ ( ( 𝐹 Fn dom 𝐹 ∧ 𝐴 ⊆ dom 𝐹 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑥 ) ∈ ( 𝐹 “ 𝐴 ) ) |
19 |
16 17 10 18
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑥 ) ∈ ( 𝐹 “ 𝐴 ) ) |
20 |
6 19
|
eqeltrrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐺 ‘ 𝑥 ) ∈ ( 𝐹 “ 𝐴 ) ) |
21 |
1 3 20
|
funimassd |
⊢ ( 𝜑 → ( 𝐺 “ 𝐴 ) ⊆ ( 𝐹 “ 𝐴 ) ) |
22 |
14 21
|
eqssd |
⊢ ( 𝜑 → ( 𝐹 “ 𝐴 ) = ( 𝐺 “ 𝐴 ) ) |