Theorem fcoreslem4

Description: Lemma 4 for fcores . (Contributed by AV, 17-Sep-2024)

	Ref	Expression
Hypotheses	fcores.f	$$⊢\phi \to F:A⟶B$$
	fcores.e	$$⊢E=\mathrm{ran}F\cap C$$
	fcores.p	$$⊢P=F^{-1}\left[C\right]$$
	fcores.x	$$⊢X={F\upharpoonright }_P$$
	fcores.g	$$⊢\phi \to G:C⟶D$$
	fcores.y	$$⊢Y={G\upharpoonright }_E$$
Assertion	fcoreslem4	$$⊢\phi \to \left(Y\circ X\right)FnP$$

Proof

Step	Hyp	Ref	Expression
1		fcores.f	$$⊢\phi \to F:A⟶B$$
2		fcores.e	$$⊢E=\mathrm{ran}F\cap C$$
3		fcores.p	$$⊢P=F^{-1}\left[C\right]$$
4		fcores.x	$$⊢X={F\upharpoonright }_P$$
5		fcores.g	$$⊢\phi \to G:C⟶D$$
6		fcores.y	$$⊢Y={G\upharpoonright }_E$$
7	5	ffnd	$$⊢\phi \to GFnC$$
8	2	a1i	$$⊢\phi \to E=\mathrm{ran}F\cap C$$
9		inss2	$$⊢\mathrm{ran}F\cap C\subseteq C$$
10	8 9	eqsstrdi	$$⊢\phi \to E\subseteq C$$
11	7 10	fnssresd	$$⊢\phi \to \left({G\upharpoonright }_E\right)FnE$$
12	6	fneq1i	$$⊢YFnE\leftrightarrow \left({G\upharpoonright }_E\right)FnE$$
13	11 12	sylibr	$$⊢\phi \to YFnE$$
14	1 2 3 4	fcoreslem3	$$⊢\phi \to X:P\underset{onto}{⟶}E$$
15		fofn	$$⊢X:P\underset{onto}{⟶}E\to XFnP$$
16	14 15	syl	$$⊢\phi \to XFnP$$
17	1 2 3 4	fcoreslem2	$$⊢\phi \to \mathrm{ran}X=E$$
18		eqimss	$$⊢\mathrm{ran}X=E\to \mathrm{ran}X\subseteq E$$
19	17 18	syl	$$⊢\phi \to \mathrm{ran}X\subseteq E$$
20		fnco	$$⊢\left(YFnE\wedge XFnP\wedge \mathrm{ran}X\subseteq E\right)\to \left(Y\circ X\right)FnP$$
21	13 16 19 20	syl3anc	$$⊢\phi \to \left(Y\circ X\right)FnP$$