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Metamath Proof Explorer

Theorem bnj1503

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

Ref Expression
Hypotheses bnj1503.1 ⊢ ( 𝜑 → Fun 𝐹 )
bnj1503.2 ⊢ ( 𝜑 → 𝐺 ⊆ 𝐹 )
bnj1503.3 ⊢ ( 𝜑 → 𝐴 ⊆ dom 𝐺 )
Assertion bnj1503 ( 𝜑 → ( 𝐹 ↾ 𝐴 ) = ( 𝐺 ↾ 𝐴 ) )

Proof

Step Hyp Ref Expression
1 bnj1503.1 ⊢ ( 𝜑 → Fun 𝐹 )
2 bnj1503.2 ⊢ ( 𝜑 → 𝐺 ⊆ 𝐹 )
3 bnj1503.3 ⊢ ( 𝜑 → 𝐴 ⊆ dom 𝐺 )
4 fun2ssres ⊢ ( ( Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝐴 ⊆ dom 𝐺 ) → ( 𝐹 ↾ 𝐴 ) = ( 𝐺 ↾ 𝐴 ) )
5 1 2 3 4 syl3anc ⊢ ( 𝜑 → ( 𝐹 ↾ 𝐴 ) = ( 𝐺 ↾ 𝐴 ) )