Theorem rabsnifsb 4095

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 4052	. . . . . . . 8 |- ( x e. { A } -> x = A )
2		sbceq1a 3342	. . . . . . . . 9 |- ( x = A -> ( ph <-> [. A / x ]. ph ) )
3	2	biimpd 207	. . . . . . . 8 |- ( x = A -> ( ph -> [. A / x ]. ph ) )
4	1, 3	syl 16	. . . . . . 7 |- ( x e. { A } -> ( ph -> [. A / x ]. ph ) )
5	4	imdistani 690	. . . . . 6 |- ( ( x e. { A } /\ ph ) -> ( x e. { A } /\ [. A / x ]. ph ) )
6	5	orcd 392	. . . . 5 |- ( ( x e. { A } /\ ph ) -> ( ( x e. { A } /\ [. A / x ]. ph ) \/ ( x e. (/) /\ -. [. A / x ]. ph ) ) )
7	2	bicomd 201	. . . . . . . . 9 |- ( x = A -> ( [. A / x ]. ph <-> ph ) )
8	1, 7	syl 16	. . . . . . . 8 |- ( x e. { A } -> ( [. A / x ]. ph <-> ph ) )
9	8	biimpd 207	. . . . . . 7 |- ( x e. { A } -> ( [. A / x ]. ph -> ph ) )
10	9	imdistani 690	. . . . . 6 |- ( ( x e. { A } /\ [. A / x ]. ph ) -> ( x e. { A } /\ ph ) )
11		noel 3789	. . . . . . . 8 |- -. x e. (/)
12	11	pm2.21i 131	. . . . . . 7 |- ( x e. (/) -> ( x e. { A } /\ ph ) )
13	12	adantr 465	. . . . . 6 |- ( ( x e. (/) /\ -. [. A / x ]. ph ) -> ( x e. { A } /\ ph ) )
14	10, 13	jaoi 379	. . . . 5 |- ( ( ( x e. { A } /\ [. A / x ]. ph ) \/ ( x e. (/) /\ -. [. A / x ]. ph ) ) -> ( x e. { A } /\ ph ) )
15	6, 14	impbii 188	. . . 4 |- ( ( x e. { A } /\ ph ) <-> ( ( x e. { A } /\ [. A / x ]. ph ) \/ ( x e. (/) /\ -. [. A / x ]. ph ) ) )
16	15	abbii 2601	. . 3 |- { x | ( x e. { A } /\ ph ) } = { x | ( ( x e. { A } /\ [. A / x ]. ph ) \/ ( x e. (/) /\ -. [. A / x ]. ph ) ) }
17		nfv 1683	. . . 4 |- F/ y ( ( x e. { A } /\ [. A / x ]. ph ) \/ ( x e. (/) /\ -. [. A / x ]. ph ) )
18		nfv 1683	. . . . . 6 |- F/ x y e. { A }
19		nfsbc1v 3351	. . . . . 6 |- F/ x [. A / x ]. ph
20	18, 19	nfan 1875	. . . . 5 |- F/ x ( y e. { A } /\ [. A / x ]. ph )
21		nfv 1683	. . . . . 6 |- F/ x y e. (/)
22	19	nfn 1849	. . . . . 6 |- F/ x -. [. A / x ]. ph
23	21, 22	nfan 1875	. . . . 5 |- F/ x ( y e. (/) /\ -. [. A / x ]. ph )
24	20, 23	nfor 1882	. . . 4 |- F/ x ( ( y e. { A } /\ [. A / x ]. ph ) \/ ( y e. (/) /\ -. [. A / x ]. ph ) )
25		eleq1 2539	. . . . . 6 |- ( x = y -> ( x e. { A } <-> y e. { A } ) )
26	25	anbi1d 704	. . . . 5 |- ( x = y -> ( ( x e. { A } /\ [. A / x ]. ph ) <-> ( y e. { A } /\ [. A / x ]. ph ) ) )
27		eleq1 2539	. . . . . 6 |- ( x = y -> ( x e. (/) <-> y e. (/) ) )
28	27	anbi1d 704	. . . . 5 |- ( x = y -> ( ( x e. (/) /\ -. [. A / x ]. ph ) <-> ( y e. (/) /\ -. [. A / x ]. ph ) ) )
29	26, 28	orbi12d 709	. . . 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 ) ) ) )
30	17, 24, 29	cbvab 2608	. . 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 ) ) }
31	16, 30	eqtri 2496	. 2 |- { x | ( x e. { A } /\ ph ) } = { y | ( ( y e. { A } /\ [. A / x ]. ph ) \/ ( y e. (/) /\ -. [. A / x ]. ph ) ) }
32		df-rab 2823	. 2 |- { x e. { A } | ph } = { x | ( x e. { A } /\ ph ) }
33		df-if 3940	. 2 |- if ( [. A / x ]. ph , { A } , (/) ) = { y | ( ( y e. { A } /\ [. A / x ]. ph ) \/ ( y e. (/) /\ -. [. A / x ]. ph ) ) }
34	31, 32, 33	3eqtr4i 2506	1 |- { x e. { A } | ph } = if ( [. A / x ]. ph , { A } , (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1379   e. wcel 1767   {cab 2452   {crab 2818   [.wsbc 3331   (/)c0 3785   ifcif 3939   {csn 4027

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-nul 3786  df-if 3940  df-sn 4028

This theorem is referenced by:  rabsnif  4096  rabrsn  4097