Metamath Proof Explorer
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)
|
|
Ref |
Expression |
|
Hypotheses |
bnj1502.1 |
⊢ ( 𝜑 → Fun 𝐹 ) |
|
|
bnj1502.2 |
⊢ ( 𝜑 → 𝐺 ⊆ 𝐹 ) |
|
|
bnj1502.3 |
⊢ ( 𝜑 → 𝐴 ∈ dom 𝐺 ) |
|
Assertion |
bnj1502 |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝐴 ) = ( 𝐺 ‘ 𝐴 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
bnj1502.1 |
⊢ ( 𝜑 → Fun 𝐹 ) |
2 |
|
bnj1502.2 |
⊢ ( 𝜑 → 𝐺 ⊆ 𝐹 ) |
3 |
|
bnj1502.3 |
⊢ ( 𝜑 → 𝐴 ∈ dom 𝐺 ) |
4 |
|
funssfv |
⊢ ( ( Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝐴 ∈ dom 𝐺 ) → ( 𝐹 ‘ 𝐴 ) = ( 𝐺 ‘ 𝐴 ) ) |
5 |
1 2 3 4
|
syl3anc |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝐴 ) = ( 𝐺 ‘ 𝐴 ) ) |