Theorem fssxp

Description: A mapping is a class of ordered pairs. (Contributed by NM, 3-Aug-1994) (Proof shortened by Andrew Salmon, 17-Sep-2011)

	Ref	Expression
Assertion	fssxp	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → 𝐹 ⊆ ( 𝐴 × 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		frel	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → Rel 𝐹 )
2		relssdmrn	⊢ ( Rel 𝐹 → 𝐹 ⊆ ( dom 𝐹 × ran 𝐹 ) )
3	1 2	syl	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → 𝐹 ⊆ ( dom 𝐹 × ran 𝐹 ) )
4		fdm	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → dom 𝐹 = 𝐴 )
5		eqimss	⊢ ( dom 𝐹 = 𝐴 → dom 𝐹 ⊆ 𝐴 )
6	4 5	syl	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → dom 𝐹 ⊆ 𝐴 )
7		frn	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → ran 𝐹 ⊆ 𝐵 )
8		xpss12	⊢ ( ( dom 𝐹 ⊆ 𝐴 ∧ ran 𝐹 ⊆ 𝐵 ) → ( dom 𝐹 × ran 𝐹 ) ⊆ ( 𝐴 × 𝐵 ) )
9	6 7 8	syl2anc	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → ( dom 𝐹 × ran 𝐹 ) ⊆ ( 𝐴 × 𝐵 ) )
10	3 9	sstrd	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → 𝐹 ⊆ ( 𝐴 × 𝐵 ) )