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Metamath Proof Explorer

Theorem bnj1502

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