Theorem rankrelp

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

	Ref	Expression
Assertion	rankrelp	Could not format assertion : No typesetting found for |- rank RelPres _E , _E ( U. ( R1 " On ) , On ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		rankf	$$⊢\mathrm{rank}:\bigcup R_1\left[\mathrm{On}\right]⟶\mathrm{On}$$
2		rankelb	$$⊢y\in \bigcup R_1\left[\mathrm{On}\right]\to \left(x\in y\to \mathrm{rank}\left(x\right)\in \mathrm{rank}\left(y\right)\right)$$
3		epel	$$⊢x\mathrm{E}y\leftrightarrow x\in y$$
4		fvex	$$⊢\mathrm{rank}\left(y\right)\in \mathrm{V}$$
5	4	epeli	$$⊢\mathrm{rank}\left(x\right)\mathrm{E}\mathrm{rank}\left(y\right)\leftrightarrow \mathrm{rank}\left(x\right)\in \mathrm{rank}\left(y\right)$$
6	2 3 5	3imtr4g	$$⊢y\in \bigcup R_1\left[\mathrm{On}\right]\to \left(x\mathrm{E}y\to \mathrm{rank}\left(x\right)\mathrm{E}\mathrm{rank}\left(y\right)\right)$$
7	6	rgen	$$⊢\forall y\in \bigcup R_1\left[\mathrm{On}\right]\left(x\mathrm{E}y\to \mathrm{rank}\left(x\right)\mathrm{E}\mathrm{rank}\left(y\right)\right)$$
8	7	rgenw	$$⊢\forall x\in \bigcup R_1\left[\mathrm{On}\right]\forall y\in \bigcup R_1\left[\mathrm{On}\right]\left(x\mathrm{E}y\to \mathrm{rank}\left(x\right)\mathrm{E}\mathrm{rank}\left(y\right)\right)$$
9		df-relp	Could not format ( rank RelPres _E , _E ( U. ( R1 " On ) , On ) <-> ( rank : U. ( R1 " On ) --> On /\ A. x e. U. ( R1 " On ) A. y e. U. ( R1 " On ) ( x _E y -> ( rank ` x ) _E ( rank ` y ) ) ) ) : No typesetting found for |- ( rank RelPres _E , _E ( U. ( R1 " On ) , On ) <-> ( rank : U. ( R1 " On ) --> On /\ A. x e. U. ( R1 " On ) A. y e. U. ( R1 " On ) ( x _E y -> ( rank ` x ) _E ( rank ` y ) ) ) ) with typecode |-
10	1 8 9	mpbir2an	Could not format rank RelPres _E , _E ( U. ( R1 " On ) , On ) : No typesetting found for |- rank RelPres _E , _E ( U. ( R1 " On ) , On ) with typecode |-