Theorem rankrelp

Description: The rank function preserves e. . (Contributed by Eric Schmidt, 11-Oct-2025)

	Ref	Expression
Assertion	rankrelp	⊢ rank RelPres E , E ( ∪ ( 𝑅1 “ On ) , On )

Proof

Step	Hyp	Ref	Expression
1		rankf	⊢ rank : ∪ ( 𝑅1 “ On ) ⟶ On
2		rankelb	⊢ ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) → ( 𝑥 ∈ 𝑦 → ( rank ‘ 𝑥 ) ∈ ( rank ‘ 𝑦 ) ) )
3		epel	⊢ ( 𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦 )
4		fvex	⊢ ( rank ‘ 𝑦 ) ∈ V
5	4	epeli	⊢ ( ( rank ‘ 𝑥 ) E ( rank ‘ 𝑦 ) ↔ ( rank ‘ 𝑥 ) ∈ ( rank ‘ 𝑦 ) )
6	2 3 5	3imtr4g	⊢ ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) → ( 𝑥 E 𝑦 → ( rank ‘ 𝑥 ) E ( rank ‘ 𝑦 ) ) )
7	6	rgen	⊢ ∀ 𝑦 ∈ ∪ ( 𝑅1 “ On ) ( 𝑥 E 𝑦 → ( rank ‘ 𝑥 ) E ( rank ‘ 𝑦 ) )
8	7	rgenw	⊢ ∀ 𝑥 ∈ ∪ ( 𝑅1 “ On ) ∀ 𝑦 ∈ ∪ ( 𝑅1 “ On ) ( 𝑥 E 𝑦 → ( rank ‘ 𝑥 ) E ( rank ‘ 𝑦 ) )
9		df-relp	⊢ ( rank RelPres E , E ( ∪ ( 𝑅1 “ On ) , On ) ↔ ( rank : ∪ ( 𝑅1 “ On ) ⟶ On ∧ ∀ 𝑥 ∈ ∪ ( 𝑅1 “ On ) ∀ 𝑦 ∈ ∪ ( 𝑅1 “ On ) ( 𝑥 E 𝑦 → ( rank ‘ 𝑥 ) E ( rank ‘ 𝑦 ) ) ) )
10	1 8 9	mpbir2an	⊢ rank RelPres E , E ( ∪ ( 𝑅1 “ On ) , On )