Theorem mptssALT

Description: Deduce subset relation of mapping-to function graphs from a subset relation of domains. Alternative proof of mptss . (Contributed by Thierry Arnoux, 30-May-2020) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	mptssALT	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ⊆ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		ssel	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) )
2	1	anim1d	⊢ ( 𝐴 ⊆ 𝐵 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶 ) → ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐶 ) ) )
3	2	ssopab2dv	⊢ ( 𝐴 ⊆ 𝐵 → { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶 ) } ⊆ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐶 ) } )
4		df-mpt	⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶 ) }
5		df-mpt	⊢ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐶 ) }
6	3 4 5	3sstr4g	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ⊆ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) )