Theorem fcoresf1lem

Description: Lemma for fcoresf1 . (Contributed by AV, 18-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	fcoresf1lem	$$⊢\left(\phi \wedge Z\in P\right)\to \left(G\circ F\right)\left(Z\right)=Y\left(X\left(Z\right)\right)$$

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	1 2 3 4 5 6	fcores	$$⊢\phi \to G\circ F=Y\circ X$$
8	7	fveq1d	$$⊢\phi \to \left(G\circ F\right)\left(Z\right)=\left(Y\circ X\right)\left(Z\right)$$
9	8	adantr	$$⊢\left(\phi \wedge Z\in P\right)\to \left(G\circ F\right)\left(Z\right)=\left(Y\circ X\right)\left(Z\right)$$
10	1 2 3 4	fcoreslem3	$$⊢\phi \to X:P\underset{onto}{⟶}E$$
11		fof	$$⊢X:P\underset{onto}{⟶}E\to X:P⟶E$$
12	10 11	syl	$$⊢\phi \to X:P⟶E$$
13	12	adantr	$$⊢\left(\phi \wedge Z\in P\right)\to X:P⟶E$$
14		simpr	$$⊢\left(\phi \wedge Z\in P\right)\to Z\in P$$
15	13 14	fvco3d	$$⊢\left(\phi \wedge Z\in P\right)\to \left(Y\circ X\right)\left(Z\right)=Y\left(X\left(Z\right)\right)$$
16	9 15	eqtrd	$$⊢\left(\phi \wedge Z\in P\right)\to \left(G\circ F\right)\left(Z\right)=Y\left(X\left(Z\right)\right)$$