Theorem elpm2r

Description: Sufficient condition for being a partial function. (Contributed by NM, 31-Dec-2013)

	Ref	Expression
Assertion	elpm2r	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) ∧ ( 𝐹 : 𝐶 ⟶ 𝐴 ∧ 𝐶 ⊆ 𝐵 ) ) → 𝐹 ∈ ( 𝐴 ↑pm 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		fdm	⊢ ( 𝐹 : 𝐶 ⟶ 𝐴 → dom 𝐹 = 𝐶 )
2	1	feq2d	⊢ ( 𝐹 : 𝐶 ⟶ 𝐴 → ( 𝐹 : dom 𝐹 ⟶ 𝐴 ↔ 𝐹 : 𝐶 ⟶ 𝐴 ) )
3	1	sseq1d	⊢ ( 𝐹 : 𝐶 ⟶ 𝐴 → ( dom 𝐹 ⊆ 𝐵 ↔ 𝐶 ⊆ 𝐵 ) )
4	2 3	anbi12d	⊢ ( 𝐹 : 𝐶 ⟶ 𝐴 → ( ( 𝐹 : dom 𝐹 ⟶ 𝐴 ∧ dom 𝐹 ⊆ 𝐵 ) ↔ ( 𝐹 : 𝐶 ⟶ 𝐴 ∧ 𝐶 ⊆ 𝐵 ) ) )
5	4	adantr	⊢ ( ( 𝐹 : 𝐶 ⟶ 𝐴 ∧ 𝐶 ⊆ 𝐵 ) → ( ( 𝐹 : dom 𝐹 ⟶ 𝐴 ∧ dom 𝐹 ⊆ 𝐵 ) ↔ ( 𝐹 : 𝐶 ⟶ 𝐴 ∧ 𝐶 ⊆ 𝐵 ) ) )
6	5	ibir	⊢ ( ( 𝐹 : 𝐶 ⟶ 𝐴 ∧ 𝐶 ⊆ 𝐵 ) → ( 𝐹 : dom 𝐹 ⟶ 𝐴 ∧ dom 𝐹 ⊆ 𝐵 ) )
7		elpm2g	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐹 ∈ ( 𝐴 ↑pm 𝐵 ) ↔ ( 𝐹 : dom 𝐹 ⟶ 𝐴 ∧ dom 𝐹 ⊆ 𝐵 ) ) )
8	6 7	imbitrrid	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ( 𝐹 : 𝐶 ⟶ 𝐴 ∧ 𝐶 ⊆ 𝐵 ) → 𝐹 ∈ ( 𝐴 ↑pm 𝐵 ) ) )
9	8	imp	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) ∧ ( 𝐹 : 𝐶 ⟶ 𝐴 ∧ 𝐶 ⊆ 𝐵 ) ) → 𝐹 ∈ ( 𝐴 ↑pm 𝐵 ) )