Theorem pjrni

Description: The range of a projection. Part of Theorem 26.2 of Halmos p. 44. (Contributed by NM, 30-Oct-1999) (Revised by Mario Carneiro, 10-Sep-2015) (New usage is discouraged.)

	Ref	Expression
Hypothesis	pjfn.1	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	pjrni	$$⊢\mathrm{ran}{\mathrm{proj}}_{ℎ}\left(H\right)=H$$

Proof

Step	Hyp	Ref	Expression
1		pjfn.1	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}$$
2	1	pjfni	$$⊢{\mathrm{proj}}_{ℎ}\left(H\right)Fn\mathscr{H}$$
3	1	pjcli	$$⊢x\in \mathscr{H}\to {\mathrm{proj}}_{ℎ}\left(H\right)\left(x\right)\in H$$
4	3	rgen	$$⊢\forall x\in \mathscr{H}{\mathrm{proj}}_{ℎ}\left(H\right)\left(x\right)\in H$$
5		ffnfv	$$⊢{\mathrm{proj}}_{ℎ}\left(H\right):\mathscr{H}⟶H\leftrightarrow \left({\mathrm{proj}}_{ℎ}\left(H\right)Fn\mathscr{H}\wedge \forall x\in \mathscr{H}{\mathrm{proj}}_{ℎ}\left(H\right)\left(x\right)\in H\right)$$
6	2 4 5	mpbir2an	$$⊢{\mathrm{proj}}_{ℎ}\left(H\right):\mathscr{H}⟶H$$
7		frn	$$⊢{\mathrm{proj}}_{ℎ}\left(H\right):\mathscr{H}⟶H\to \mathrm{ran}{\mathrm{proj}}_{ℎ}\left(H\right)\subseteq H$$
8	6 7	ax-mp	$$⊢\mathrm{ran}{\mathrm{proj}}_{ℎ}\left(H\right)\subseteq H$$
9		pjid	$$⊢\left(H\in {\mathbf{C}}_{\mathscr{H}}\wedge y\in H\right)\to {\mathrm{proj}}_{ℎ}\left(H\right)\left(y\right)=y$$
10	1 9	mpan	$$⊢y\in H\to {\mathrm{proj}}_{ℎ}\left(H\right)\left(y\right)=y$$
11	1	cheli	$$⊢y\in H\to y\in \mathscr{H}$$
12		fnfvelrn	$$⊢\left({\mathrm{proj}}_{ℎ}\left(H\right)Fn\mathscr{H}\wedge y\in \mathscr{H}\right)\to {\mathrm{proj}}_{ℎ}\left(H\right)\left(y\right)\in \mathrm{ran}{\mathrm{proj}}_{ℎ}\left(H\right)$$
13	2 11 12	sylancr	$$⊢y\in H\to {\mathrm{proj}}_{ℎ}\left(H\right)\left(y\right)\in \mathrm{ran}{\mathrm{proj}}_{ℎ}\left(H\right)$$
14	10 13	eqeltrrd	$$⊢y\in H\to y\in \mathrm{ran}{\mathrm{proj}}_{ℎ}\left(H\right)$$
15	14	ssriv	$$⊢H\subseteq \mathrm{ran}{\mathrm{proj}}_{ℎ}\left(H\right)$$
16	8 15	eqssi	$$⊢\mathrm{ran}{\mathrm{proj}}_{ℎ}\left(H\right)=H$$