Theorem fcoresf1b

Description: A composition is injective iff the restrictions of its components to the minimum domains are injective. (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	fcoresf1b	$$⊢\phi \to \left(\left(G\circ F\right):P\underset{1-1}{⟶}D\leftrightarrow \left(X:P\underset{1-1}{⟶}E\wedge Y:E\underset{1-1}{⟶}D\right)\right)$$

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	adantr	$$⊢\left(\phi \wedge \left(G\circ F\right):P\underset{1-1}{⟶}D\right)\to F:A⟶B$$
8	5	adantr	$$⊢\left(\phi \wedge \left(G\circ F\right):P\underset{1-1}{⟶}D\right)\to G:C⟶D$$
9		simpr	$$⊢\left(\phi \wedge \left(G\circ F\right):P\underset{1-1}{⟶}D\right)\to \left(G\circ F\right):P\underset{1-1}{⟶}D$$
10	7 2 3 4 8 6 9	fcoresf1	$$⊢\left(\phi \wedge \left(G\circ F\right):P\underset{1-1}{⟶}D\right)\to \left(X:P\underset{1-1}{⟶}E\wedge Y:E\underset{1-1}{⟶}D\right)$$
11	10	ex	$$⊢\phi \to \left(\left(G\circ F\right):P\underset{1-1}{⟶}D\to \left(X:P\underset{1-1}{⟶}E\wedge Y:E\underset{1-1}{⟶}D\right)\right)$$
12		f1co	$$⊢\left(Y:E\underset{1-1}{⟶}D\wedge X:P\underset{1-1}{⟶}E\right)\to \left(Y\circ X\right):P\underset{1-1}{⟶}D$$
13	12	ancoms	$$⊢\left(X:P\underset{1-1}{⟶}E\wedge Y:E\underset{1-1}{⟶}D\right)\to \left(Y\circ X\right):P\underset{1-1}{⟶}D$$
14	1 2 3 4 5 6	fcores	$$⊢\phi \to G\circ F=Y\circ X$$
15		f1eq1	$$⊢G\circ F=Y\circ X\to \left(\left(G\circ F\right):P\underset{1-1}{⟶}D\leftrightarrow \left(Y\circ X\right):P\underset{1-1}{⟶}D\right)$$
16	14 15	syl	$$⊢\phi \to \left(\left(G\circ F\right):P\underset{1-1}{⟶}D\leftrightarrow \left(Y\circ X\right):P\underset{1-1}{⟶}D\right)$$
17	13 16	syl5ibr	$$⊢\phi \to \left(\left(X:P\underset{1-1}{⟶}E\wedge Y:E\underset{1-1}{⟶}D\right)\to \left(G\circ F\right):P\underset{1-1}{⟶}D\right)$$
18	11 17	impbid	$$⊢\phi \to \left(\left(G\circ F\right):P\underset{1-1}{⟶}D\leftrightarrow \left(X:P\underset{1-1}{⟶}E\wedge Y:E\underset{1-1}{⟶}D\right)\right)$$