Theorem psrbaglecl

Description: The set of finite bags is downward-closed. (Contributed by Mario Carneiro, 29-Dec-2014)

	Ref	Expression
Hypothesis	psrbag.d	⊢ 𝐷 = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin }
Assertion	psrbaglecl	⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → 𝐺 ∈ 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		psrbag.d	⊢ 𝐷 = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin }
2		simpr2	⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → 𝐺 : 𝐼 ⟶ ℕ0 )
3		simpr1	⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → 𝐹 ∈ 𝐷 )
4	1	psrbag	⊢ ( 𝐼 ∈ 𝑉 → ( 𝐹 ∈ 𝐷 ↔ ( 𝐹 : 𝐼 ⟶ ℕ0 ∧ ( ◡ 𝐹 “ ℕ ) ∈ Fin ) ) )
5	4	adantr	⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → ( 𝐹 ∈ 𝐷 ↔ ( 𝐹 : 𝐼 ⟶ ℕ0 ∧ ( ◡ 𝐹 “ ℕ ) ∈ Fin ) ) )
6	3 5	mpbid	⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → ( 𝐹 : 𝐼 ⟶ ℕ0 ∧ ( ◡ 𝐹 “ ℕ ) ∈ Fin ) )
7	6	simprd	⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → ( ◡ 𝐹 “ ℕ ) ∈ Fin )
8	1	psrbaglesupp	⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → ( ◡ 𝐺 “ ℕ ) ⊆ ( ◡ 𝐹 “ ℕ ) )
9	7 8	ssfid	⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → ( ◡ 𝐺 “ ℕ ) ∈ Fin )
10	1	psrbag	⊢ ( 𝐼 ∈ 𝑉 → ( 𝐺 ∈ 𝐷 ↔ ( 𝐺 : 𝐼 ⟶ ℕ0 ∧ ( ◡ 𝐺 “ ℕ ) ∈ Fin ) ) )
11	10	adantr	⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → ( 𝐺 ∈ 𝐷 ↔ ( 𝐺 : 𝐼 ⟶ ℕ0 ∧ ( ◡ 𝐺 “ ℕ ) ∈ Fin ) ) )
12	2 9 11	mpbir2and	⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝐺 : 𝐼 ⟶ ℕ0 ∧ 𝐺 ∘r ≤ 𝐹 ) ) → 𝐺 ∈ 𝐷 )