Theorem dff2

Description: Alternate definition of a mapping. (Contributed by NM, 14-Nov-2007)

	Ref	Expression
Assertion	dff2	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 ↔ ( 𝐹 Fn 𝐴 ∧ 𝐹 ⊆ ( 𝐴 × 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ffn	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → 𝐹 Fn 𝐴 )
2		fssxp	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → 𝐹 ⊆ ( 𝐴 × 𝐵 ) )
3	1 2	jca	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → ( 𝐹 Fn 𝐴 ∧ 𝐹 ⊆ ( 𝐴 × 𝐵 ) ) )
4		rnss	⊢ ( 𝐹 ⊆ ( 𝐴 × 𝐵 ) → ran 𝐹 ⊆ ran ( 𝐴 × 𝐵 ) )
5		rnxpss	⊢ ran ( 𝐴 × 𝐵 ) ⊆ 𝐵
6	4 5	sstrdi	⊢ ( 𝐹 ⊆ ( 𝐴 × 𝐵 ) → ran 𝐹 ⊆ 𝐵 )
7	6	anim2i	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐹 ⊆ ( 𝐴 × 𝐵 ) ) → ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵 ) )
8		df-f	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 ↔ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵 ) )
9	7 8	sylibr	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐹 ⊆ ( 𝐴 × 𝐵 ) ) → 𝐹 : 𝐴 ⟶ 𝐵 )
10	3 9	impbii	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 ↔ ( 𝐹 Fn 𝐴 ∧ 𝐹 ⊆ ( 𝐴 × 𝐵 ) ) )