Theorem fcoresfo

Description: If a composition is surjective, then the restriction of its first component to the minimum domain is surjective. (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	$$⊢Y={G\upharpoonright }_E$$
	fcoresfo.s	$$⊢\phi \to \left(G\circ F\right):P\underset{onto}{⟶}D$$
Assertion	fcoresfo	$$⊢\phi \to Y:E\underset{onto}{⟶}D$$

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		fcoresfo.s	$$⊢\phi \to \left(G\circ F\right):P\underset{onto}{⟶}D$$
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	5 10	fssresd	$$⊢\phi \to \left({G\upharpoonright }_E\right):E⟶D$$
12	6	feq1i	$$⊢Y:E⟶D\leftrightarrow \left({G\upharpoonright }_E\right):E⟶D$$
13	11 12	sylibr	$$⊢\phi \to Y:E⟶D$$
14	1 2 3 4	fcoreslem3	$$⊢\phi \to X:P\underset{onto}{⟶}E$$
15		fof	$$⊢X:P\underset{onto}{⟶}E\to X:P⟶E$$
16	14 15	syl	$$⊢\phi \to X:P⟶E$$
17	1 2 3 4 5 6	fcores	$$⊢\phi \to G\circ F=Y\circ X$$
18	17	eqcomd	$$⊢\phi \to Y\circ X=G\circ F$$
19		foeq1	$$⊢Y\circ X=G\circ F\to \left(\left(Y\circ X\right):P\underset{onto}{⟶}D\leftrightarrow \left(G\circ F\right):P\underset{onto}{⟶}D\right)$$
20	18 19	syl	$$⊢\phi \to \left(\left(Y\circ X\right):P\underset{onto}{⟶}D\leftrightarrow \left(G\circ F\right):P\underset{onto}{⟶}D\right)$$
21	7 20	mpbird	$$⊢\phi \to \left(Y\circ X\right):P\underset{onto}{⟶}D$$
22		foco2	$$⊢\left(Y:E⟶D\wedge X:P⟶E\wedge \left(Y\circ X\right):P\underset{onto}{⟶}D\right)\to Y:E\underset{onto}{⟶}D$$
23	13 16 21 22	syl3anc	$$⊢\phi \to Y:E\underset{onto}{⟶}D$$