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Metamath Proof Explorer

Theorem bnj1422

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

Ref Expression
Hypotheses bnj1422.1 ⊢ ( 𝜑 → Fun 𝐴 )
bnj1422.2 ⊢ ( 𝜑 → dom 𝐴 = 𝐵 )
Assertion bnj1422 ( 𝜑 → 𝐴 Fn 𝐵 )

Proof

Step Hyp Ref Expression
1 bnj1422.1 ⊢ ( 𝜑 → Fun 𝐴 )
2 bnj1422.2 ⊢ ( 𝜑 → dom 𝐴 = 𝐵 )
3 df-fn ⊢ ( 𝐴 Fn 𝐵 ↔ ( Fun 𝐴 ∧ dom 𝐴 = 𝐵 ) )
4 1 2 3 sylanbrc ⊢ ( 𝜑 → 𝐴 Fn 𝐵 )