Theorem frege129

Description: If the procedure R is single-valued and Y belongs to the R -sequence beginning with M or precedes M in the R -sequence, then every result of an application of the procedure R to Y belongs to the R -sequence beginning with M or precedes M in the R -sequence. Proposition 129 of Frege1879 p. 83. (Contributed by RP, 9-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege123.x	$$⊢X\in U$$
	frege123.y	$$⊢Y\in V$$
	frege124.m	$$⊢M\in W$$
	frege124.r	$$⊢R\in S$$
Assertion	frege129	$$⊢\mathrm{Fun}{R^{-1}}^{-1}\to \left(\left(\neg Y\mathrm{t+}\left(R\right)M\to M\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Y\right)\to \left(YRX\to \left(\neg X\mathrm{t+}\left(R\right)M\to M\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)X\right)\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege123.x	$$⊢X\in U$$
2		frege123.y	$$⊢Y\in V$$
3		frege124.m	$$⊢M\in W$$
4		frege124.r	$$⊢R\in S$$
5	3 2 1 4	frege111	$$⊢M\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Y\to \left(YRX\to \left(\neg X\mathrm{t+}\left(R\right)M\to M\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)X\right)\right)$$
6	1 2 3 4	frege128	$$⊢\left(M\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Y\to \left(YRX\to \left(\neg X\mathrm{t+}\left(R\right)M\to M\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)X\right)\right)\right)\to \left(\mathrm{Fun}{R^{-1}}^{-1}\to \left(\left(\neg Y\mathrm{t+}\left(R\right)M\to M\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Y\right)\to \left(YRX\to \left(\neg X\mathrm{t+}\left(R\right)M\to M\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)X\right)\right)\right)\right)$$
7	5 6	ax-mp	$$⊢\mathrm{Fun}{R^{-1}}^{-1}\to \left(\left(\neg Y\mathrm{t+}\left(R\right)M\to M\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)Y\right)\to \left(YRX\to \left(\neg X\mathrm{t+}\left(R\right)M\to M\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)X\right)\right)\right)$$