Theorem funressnmo

Description: A function restricted to a singleton has at most one value for the singleton element as argument. (Contributed by AV, 2-Sep-2022)

	Ref	Expression
Assertion	funressnmo	$$⊢\left(A\in V\wedge \mathrm{Fun}\left({F\upharpoonright }_{\left\{A\right\}}\right)\right)\to {\exists }^{*}yAFy$$

Proof

Step	Hyp	Ref	Expression
1		sneq	$$⊢x=A\to \left\{x\right\}=\left\{A\right\}$$
2	1	reseq2d	$$⊢x=A\to {F\upharpoonright }_{\left\{x\right\}}={F\upharpoonright }_{\left\{A\right\}}$$
3	2	funeqd	$$⊢x=A\to \left(\mathrm{Fun}\left({F\upharpoonright }_{\left\{x\right\}}\right)\leftrightarrow \mathrm{Fun}\left({F\upharpoonright }_{\left\{A\right\}}\right)\right)$$
4		breq1	$$⊢x=A\to \left(xFy\leftrightarrow AFy\right)$$
5	4	mobidv	$$⊢x=A\to \left({\exists }^{*}yxFy\leftrightarrow {\exists }^{*}yAFy\right)$$
6	3 5	imbi12d	$$⊢x=A\to \left(\left(\mathrm{Fun}\left({F\upharpoonright }_{\left\{x\right\}}\right)\to {\exists }^{*}yxFy\right)\leftrightarrow \left(\mathrm{Fun}\left({F\upharpoonright }_{\left\{A\right\}}\right)\to {\exists }^{*}yAFy\right)\right)$$
7		funressnvmo	$$⊢\mathrm{Fun}\left({F\upharpoonright }_{\left\{x\right\}}\right)\to {\exists }^{*}yxFy$$
8	6 7	vtoclg	$$⊢A\in V\to \left(\mathrm{Fun}\left({F\upharpoonright }_{\left\{A\right\}}\right)\to {\exists }^{*}yAFy\right)$$
9	8	imp	$$⊢\left(A\in V\wedge \mathrm{Fun}\left({F\upharpoonright }_{\left\{A\right\}}\right)\right)\to {\exists }^{*}yAFy$$