Theorem fcoresfob

Description: A composition is surjective iff the restriction of its first component to the minimum domain is surjective. (Contributed by GL and AV, 7-Oct-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	fcoresfob	$$⊢\phi \to \left(\left(G\circ F\right):P\underset{onto}{⟶}D\leftrightarrow Y:E\underset{onto}{⟶}D\right)$$

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	adantr	$$⊢\left(\phi \wedge \left(G\circ F\right):P\underset{onto}{⟶}D\right)\to F:A⟶B$$
8	5	adantr	$$⊢\left(\phi \wedge \left(G\circ F\right):P\underset{onto}{⟶}D\right)\to G:C⟶D$$
9		simpr	$$⊢\left(\phi \wedge \left(G\circ F\right):P\underset{onto}{⟶}D\right)\to \left(G\circ F\right):P\underset{onto}{⟶}D$$
10	7 2 3 4 8 6 9	fcoresfo	$$⊢\left(\phi \wedge \left(G\circ F\right):P\underset{onto}{⟶}D\right)\to Y:E\underset{onto}{⟶}D$$
11	1 2 3 4	fcoreslem3	$$⊢\phi \to X:P\underset{onto}{⟶}E$$
12	11	anim1ci	$$⊢\left(\phi \wedge Y:E\underset{onto}{⟶}D\right)\to \left(Y:E\underset{onto}{⟶}D\wedge X:P\underset{onto}{⟶}E\right)$$
13		foco	$$⊢\left(Y:E\underset{onto}{⟶}D\wedge X:P\underset{onto}{⟶}E\right)\to \left(Y\circ X\right):P\underset{onto}{⟶}D$$
14	12 13	syl	$$⊢\left(\phi \wedge Y:E\underset{onto}{⟶}D\right)\to \left(Y\circ X\right):P\underset{onto}{⟶}D$$
15	1 2 3 4 5 6	fcores	$$⊢\phi \to G\circ F=Y\circ X$$
16	15	adantr	$$⊢\left(\phi \wedge Y:E\underset{onto}{⟶}D\right)\to G\circ F=Y\circ X$$
17		foeq1	$$⊢G\circ F=Y\circ X\to \left(\left(G\circ F\right):P\underset{onto}{⟶}D\leftrightarrow \left(Y\circ X\right):P\underset{onto}{⟶}D\right)$$
18	16 17	syl	$$⊢\left(\phi \wedge Y:E\underset{onto}{⟶}D\right)\to \left(\left(G\circ F\right):P\underset{onto}{⟶}D\leftrightarrow \left(Y\circ X\right):P\underset{onto}{⟶}D\right)$$
19	14 18	mpbird	$$⊢\left(\phi \wedge Y:E\underset{onto}{⟶}D\right)\to \left(G\circ F\right):P\underset{onto}{⟶}D$$
20	10 19	impbida	$$⊢\phi \to \left(\left(G\circ F\right):P\underset{onto}{⟶}D\leftrightarrow Y:E\underset{onto}{⟶}D\right)$$