Theorem ffdm

Description: A mapping is a partial function. (Contributed by NM, 25-Nov-2007)

	Ref	Expression
Assertion	ffdm	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → ( 𝐹 : dom 𝐹 ⟶ 𝐵 ∧ dom 𝐹 ⊆ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		fdm	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → dom 𝐹 = 𝐴 )
2	1	feq2d	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → ( 𝐹 : dom 𝐹 ⟶ 𝐵 ↔ 𝐹 : 𝐴 ⟶ 𝐵 ) )
3	2	ibir	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → 𝐹 : dom 𝐹 ⟶ 𝐵 )
4		eqimss	⊢ ( dom 𝐹 = 𝐴 → dom 𝐹 ⊆ 𝐴 )
5	1 4	syl	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → dom 𝐹 ⊆ 𝐴 )
6	3 5	jca	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → ( 𝐹 : dom 𝐹 ⟶ 𝐵 ∧ dom 𝐹 ⊆ 𝐴 ) )