Theorem rankelg

Description: The membership relation is inherited by the rank function. Closed form of rankel . (Contributed by Scott Fenton, 16-Jul-2015)

	Ref	Expression
Assertion	rankelg	⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝐵 ) → ( rank ‘ 𝐴 ) ∈ ( rank ‘ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		eleq2	⊢ ( 𝑦 = 𝐵 → ( 𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝐵 ) )
2		fveq2	⊢ ( 𝑦 = 𝐵 → ( rank ‘ 𝑦 ) = ( rank ‘ 𝐵 ) )
3	2	eleq2d	⊢ ( 𝑦 = 𝐵 → ( ( rank ‘ 𝐴 ) ∈ ( rank ‘ 𝑦 ) ↔ ( rank ‘ 𝐴 ) ∈ ( rank ‘ 𝐵 ) ) )
4	1 3	imbi12d	⊢ ( 𝑦 = 𝐵 → ( ( 𝐴 ∈ 𝑦 → ( rank ‘ 𝐴 ) ∈ ( rank ‘ 𝑦 ) ) ↔ ( 𝐴 ∈ 𝐵 → ( rank ‘ 𝐴 ) ∈ ( rank ‘ 𝐵 ) ) ) )
5		vex	⊢ 𝑦 ∈ V
6	5	rankel	⊢ ( 𝐴 ∈ 𝑦 → ( rank ‘ 𝐴 ) ∈ ( rank ‘ 𝑦 ) )
7	4 6	vtoclg	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐴 ∈ 𝐵 → ( rank ‘ 𝐴 ) ∈ ( rank ‘ 𝐵 ) ) )
8	7	imp	⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝐵 ) → ( rank ‘ 𝐴 ) ∈ ( rank ‘ 𝐵 ) )