Theorem elorvc

Description: Elementhood of a preimage. (Contributed by Thierry Arnoux, 21-Jan-2017)

	Ref	Expression
Hypotheses	orvcval.1	⊢ ( 𝜑 → Fun 𝑋 )
	orvcval.2	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
	orvcval.3	⊢ ( 𝜑 → 𝐴 ∈ 𝑊 )
Assertion	elorvc	⊢ ( ( 𝜑 ∧ 𝑧 ∈ dom 𝑋 ) → ( 𝑧 ∈ ( 𝑋 ∘RV/𝑐 𝑅 𝐴 ) ↔ ( 𝑋 ‘ 𝑧 ) 𝑅 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		orvcval.1	⊢ ( 𝜑 → Fun 𝑋 )
2		orvcval.2	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
3		orvcval.3	⊢ ( 𝜑 → 𝐴 ∈ 𝑊 )
4	1 2 3	orvcval2	⊢ ( 𝜑 → ( 𝑋 ∘RV/𝑐 𝑅 𝐴 ) = { 𝑧 ∈ dom 𝑋 ∣ ( 𝑋 ‘ 𝑧 ) 𝑅 𝐴 } )
5	4	eleq2d	⊢ ( 𝜑 → ( 𝑧 ∈ ( 𝑋 ∘RV/𝑐 𝑅 𝐴 ) ↔ 𝑧 ∈ { 𝑧 ∈ dom 𝑋 ∣ ( 𝑋 ‘ 𝑧 ) 𝑅 𝐴 } ) )
6		rabid	⊢ ( 𝑧 ∈ { 𝑧 ∈ dom 𝑋 ∣ ( 𝑋 ‘ 𝑧 ) 𝑅 𝐴 } ↔ ( 𝑧 ∈ dom 𝑋 ∧ ( 𝑋 ‘ 𝑧 ) 𝑅 𝐴 ) )
7	5 6	bitrdi	⊢ ( 𝜑 → ( 𝑧 ∈ ( 𝑋 ∘RV/𝑐 𝑅 𝐴 ) ↔ ( 𝑧 ∈ dom 𝑋 ∧ ( 𝑋 ‘ 𝑧 ) 𝑅 𝐴 ) ) )
8	7	baibd	⊢ ( ( 𝜑 ∧ 𝑧 ∈ dom 𝑋 ) → ( 𝑧 ∈ ( 𝑋 ∘RV/𝑐 𝑅 𝐴 ) ↔ ( 𝑋 ‘ 𝑧 ) 𝑅 𝐴 ) )