Theorem funconstss

Description: Two ways of specifying that a function is constant on a subdomain. (Contributed by NM, 8-Mar-2007)

	Ref	Expression
Assertion	funconstss	⊢ ( ( Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹 ) → ( ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝐵 ↔ 𝐴 ⊆ ( ◡ 𝐹 “ { 𝐵 } ) ) )

Proof

Step	Hyp	Ref	Expression
1		funimass4	⊢ ( ( Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹 ) → ( ( 𝐹 “ 𝐴 ) ⊆ { 𝐵 } ↔ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ∈ { 𝐵 } ) )
2		fvex	⊢ ( 𝐹 ‘ 𝑥 ) ∈ V
3	2	elsn	⊢ ( ( 𝐹 ‘ 𝑥 ) ∈ { 𝐵 } ↔ ( 𝐹 ‘ 𝑥 ) = 𝐵 )
4	3	ralbii	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ∈ { 𝐵 } ↔ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝐵 )
5	1 4	bitr2di	⊢ ( ( Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹 ) → ( ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝐵 ↔ ( 𝐹 “ 𝐴 ) ⊆ { 𝐵 } ) )
6		funimass3	⊢ ( ( Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹 ) → ( ( 𝐹 “ 𝐴 ) ⊆ { 𝐵 } ↔ 𝐴 ⊆ ( ◡ 𝐹 “ { 𝐵 } ) ) )
7	5 6	bitrd	⊢ ( ( Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹 ) → ( ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝐵 ↔ 𝐴 ⊆ ( ◡ 𝐹 “ { 𝐵 } ) ) )