Theorem frege121

Description: Lemma for frege122 . Proposition 121 of Frege1879 p. 79. (Contributed by RP, 8-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege116.x	$$⊢X\in U$$
	frege118.y	$$⊢Y\in V$$
	frege120.a	$$⊢A\in W$$
Assertion	frege121	$$⊢\left(A=X\to X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)A\right)\to \left(\mathrm{Fun}{R^{-1}}^{-1}\to \left(YRX\to \left(YRA\to X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)A\right)\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege116.x	$$⊢X\in U$$
2		frege118.y	$$⊢Y\in V$$
3		frege120.a	$$⊢A\in W$$
4	1 2 3	frege120	$$⊢\mathrm{Fun}{R^{-1}}^{-1}\to \left(YRX\to \left(YRA\to A=X\right)\right)$$
5		frege20	$$⊢\left(\mathrm{Fun}{R^{-1}}^{-1}\to \left(YRX\to \left(YRA\to A=X\right)\right)\right)\to \left(\left(A=X\to X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)A\right)\to \left(\mathrm{Fun}{R^{-1}}^{-1}\to \left(YRX\to \left(YRA\to X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)A\right)\right)\right)\right)$$
6	4 5	ax-mp	$$⊢\left(A=X\to X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)A\right)\to \left(\mathrm{Fun}{R^{-1}}^{-1}\to \left(YRX\to \left(YRA\to X\left(\mathrm{t+}\left(R\right)\cup \mathrm{I}\right)A\right)\right)\right)$$