Theorem frege123

Description: Lemma for frege124 . Proposition 123 of Frege1879 p. 79. (Contributed by RP, 8-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege123.x	$$⊢X\in U$$
	frege123.y	$$⊢Y\in V$$
Assertion	frege123	$$⊢\left(\forall a\left(YRa\to X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)a\right)\to \left(Y\mathrm{t+}\left(R\right)M\to X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)M\right)\right)\to \left(\mathrm{Fun}{R^{-1}}^{-1}\to \left(YRX\to \left(Y\mathrm{t+}\left(R\right)M\to X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)M\right)\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege123.x	$$⊢X\in U$$
2		frege123.y	$$⊢Y\in V$$
3		vex	$$⊢a\in \mathrm{V}$$
4	1 2 3	frege122	$$⊢\mathrm{Fun}{R^{-1}}^{-1}\to \left(YRX\to \left(YRa\to X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)a\right)\right)$$
5	4	alrimdv	$$⊢\mathrm{Fun}{R^{-1}}^{-1}\to \left(YRX\to \forall a\left(YRa\to X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)a\right)\right)$$
6		frege19	$$⊢\left(\mathrm{Fun}{R^{-1}}^{-1}\to \left(YRX\to \forall a\left(YRa\to X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)a\right)\right)\right)\to \left(\left(\forall a\left(YRa\to X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)a\right)\to \left(Y\mathrm{t+}\left(R\right)M\to X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)M\right)\right)\to \left(\mathrm{Fun}{R^{-1}}^{-1}\to \left(YRX\to \left(Y\mathrm{t+}\left(R\right)M\to X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)M\right)\right)\right)\right)$$
7	5 6	ax-mp	$$⊢\left(\forall a\left(YRa\to X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)a\right)\to \left(Y\mathrm{t+}\left(R\right)M\to X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)M\right)\right)\to \left(\mathrm{Fun}{R^{-1}}^{-1}\to \left(YRX\to \left(Y\mathrm{t+}\left(R\right)M\to X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)M\right)\right)\right)$$