Theorem rabsnifsb 3737

Description: A restricted class abstraction restricted to a singleton is either the empty set or the singleton itself. (Contributed by AV, 21-Jul-2019.)

Assertion

Ref	Expression
rabsnifsb	|- { x e. { A } | ph } = if ( [. A / x ]. ph , { A } , (/) )

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem rabsnifsb

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elsni 3687	. . . . . . . 8 |- ( x e. { A } -> x = A )
2		sbceq1a 3041	. . . . . . . . 9 |- ( x = A -> ( ph <-> [. A / x ]. ph ) )
3	2	biimpd 144	. . . . . . . 8 |- ( x = A -> ( ph -> [. A / x ]. ph ) )
4	1, 3	syl 14	. . . . . . 7 |- ( x e. { A } -> ( ph -> [. A / x ]. ph ) )
5	4	imdistani 445	. . . . . 6 |- ( ( x e. { A } /\ ph ) -> ( x e. { A } /\ [. A / x ]. ph ) )
6	5	orcd 740	. . . . 5 |- ( ( x e. { A } /\ ph ) -> ( ( x e. { A } /\ [. A / x ]. ph ) \/ ( x e. (/) /\ -. [. A / x ]. ph ) ) )
7	2	biimprd 158	. . . . . . . 8 |- ( x = A -> ( [. A / x ]. ph -> ph ) )
8	1, 7	syl 14	. . . . . . 7 |- ( x e. { A } -> ( [. A / x ]. ph -> ph ) )
9	8	imdistani 445	. . . . . 6 |- ( ( x e. { A } /\ [. A / x ]. ph ) -> ( x e. { A } /\ ph ) )
10		noel 3498	. . . . . . . 8 |- -. x e. (/)
11	10	pm2.21i 651	. . . . . . 7 |- ( x e. (/) -> ( x e. { A } /\ ph ) )
12	11	adantr 276	. . . . . 6 |- ( ( x e. (/) /\ -. [. A / x ]. ph ) -> ( x e. { A } /\ ph ) )
13	9, 12	jaoi 723	. . . . 5 |- ( ( ( x e. { A } /\ [. A / x ]. ph ) \/ ( x e. (/) /\ -. [. A / x ]. ph ) ) -> ( x e. { A } /\ ph ) )
14	6, 13	impbii 126	. . . 4 |- ( ( x e. { A } /\ ph ) <-> ( ( x e. { A } /\ [. A / x ]. ph ) \/ ( x e. (/) /\ -. [. A / x ]. ph ) ) )
15	14	abbii 2347	. . 3 |- { x | ( x e. { A } /\ ph ) } = { x | ( ( x e. { A } /\ [. A / x ]. ph ) \/ ( x e. (/) /\ -. [. A / x ]. ph ) ) }
16		nfv 1576	. . . 4 |- F/ y ( ( x e. { A } /\ [. A / x ]. ph ) \/ ( x e. (/) /\ -. [. A / x ]. ph ) )
17		nfv 1576	. . . . . 6 |- F/ x y e. { A }
18		nfsbc1v 3050	. . . . . 6 |- F/ x [. A / x ]. ph
19	17, 18	nfan 1613	. . . . 5 |- F/ x ( y e. { A } /\ [. A / x ]. ph )
20		nfv 1576	. . . . . 6 |- F/ x y e. (/)
21	18	nfn 1706	. . . . . 6 |- F/ x -. [. A / x ]. ph
22	20, 21	nfan 1613	. . . . 5 |- F/ x ( y e. (/) /\ -. [. A / x ]. ph )
23	19, 22	nfor 1622	. . . 4 |- F/ x ( ( y e. { A } /\ [. A / x ]. ph ) \/ ( y e. (/) /\ -. [. A / x ]. ph ) )
24		eleq1w 2292	. . . . . 6 |- ( x = y -> ( x e. { A } <-> y e. { A } ) )
25	24	anbi1d 465	. . . . 5 |- ( x = y -> ( ( x e. { A } /\ [. A / x ]. ph ) <-> ( y e. { A } /\ [. A / x ]. ph ) ) )
26		eleq1w 2292	. . . . . 6 |- ( x = y -> ( x e. (/) <-> y e. (/) ) )
27	26	anbi1d 465	. . . . 5 |- ( x = y -> ( ( x e. (/) /\ -. [. A / x ]. ph ) <-> ( y e. (/) /\ -. [. A / x ]. ph ) ) )
28	25, 27	orbi12d 800	. . . 4 |- ( x = y -> ( ( ( x e. { A } /\ [. A / x ]. ph ) \/ ( x e. (/) /\ -. [. A / x ]. ph ) ) <-> ( ( y e. { A } /\ [. A / x ]. ph ) \/ ( y e. (/) /\ -. [. A / x ]. ph ) ) ) )
29	16, 23, 28	cbvabw 2354	. . 3 |- { x | ( ( x e. { A } /\ [. A / x ]. ph ) \/ ( x e. (/) /\ -. [. A / x ]. ph ) ) } = { y | ( ( y e. { A } /\ [. A / x ]. ph ) \/ ( y e. (/) /\ -. [. A / x ]. ph ) ) }
30	15, 29	eqtri 2252	. 2 |- { x | ( x e. { A } /\ ph ) } = { y | ( ( y e. { A } /\ [. A / x ]. ph ) \/ ( y e. (/) /\ -. [. A / x ]. ph ) ) }
31		df-rab 2519	. 2 |- { x e. { A } | ph } = { x | ( x e. { A } /\ ph ) }
32		df-if 3606	. 2 |- if ( [. A / x ]. ph , { A } , (/) ) = { y | ( ( y e. { A } /\ [. A / x ]. ph ) \/ ( y e. (/) /\ -. [. A / x ]. ph ) ) }
33	30, 31, 32	3eqtr4i 2262	1 |- { x e. { A } | ph } = if ( [. A / x ]. ph , { A } , (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 715   = wceq 1397   e. wcel 2202   {cab 2217   {crab 2514   [.wsbc 3031   (/)c0 3494   ifcif 3605   {csn 3669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-nul 3495  df-if 3606  df-sn 3675

This theorem is referenced by:  rabsnif  3738