Theorem rabsnifsb 4257

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 4194	. . . . . . . 8 |- ( x e. { A } -> x = A )
2		sbceq1a 3446	. . . . . . . . 9 |- ( x = A -> ( ph <-> [. A / x ]. ph ) )
3	2	biimpd 219	. . . . . . . 8 |- ( x = A -> ( ph -> [. A / x ]. ph ) )
4	1, 3	syl 17	. . . . . . 7 |- ( x e. { A } -> ( ph -> [. A / x ]. ph ) )
5	4	imdistani 726	. . . . . 6 |- ( ( x e. { A } /\ ph ) -> ( x e. { A } /\ [. A / x ]. ph ) )
6	5	orcd 407	. . . . 5 |- ( ( x e. { A } /\ ph ) -> ( ( x e. { A } /\ [. A / x ]. ph ) \/ ( x e. (/) /\ -. [. A / x ]. ph ) ) )
7	2	biimprd 238	. . . . . . . 8 |- ( x = A -> ( [. A / x ]. ph -> ph ) )
8	1, 7	syl 17	. . . . . . 7 |- ( x e. { A } -> ( [. A / x ]. ph -> ph ) )
9	8	imdistani 726	. . . . . 6 |- ( ( x e. { A } /\ [. A / x ]. ph ) -> ( x e. { A } /\ ph ) )
10		noel 3919	. . . . . . . 8 |- -. x e. (/)
11	10	pm2.21i 116	. . . . . . 7 |- ( x e. (/) -> ( x e. { A } /\ ph ) )
12	11	adantr 481	. . . . . 6 |- ( ( x e. (/) /\ -. [. A / x ]. ph ) -> ( x e. { A } /\ ph ) )
13	9, 12	jaoi 394	. . . . 5 |- ( ( ( x e. { A } /\ [. A / x ]. ph ) \/ ( x e. (/) /\ -. [. A / x ]. ph ) ) -> ( x e. { A } /\ ph ) )
14	6, 13	impbii 199	. . . 4 |- ( ( x e. { A } /\ ph ) <-> ( ( x e. { A } /\ [. A / x ]. ph ) \/ ( x e. (/) /\ -. [. A / x ]. ph ) ) )
15	14	abbii 2739	. . 3 |- { x | ( x e. { A } /\ ph ) } = { x | ( ( x e. { A } /\ [. A / x ]. ph ) \/ ( x e. (/) /\ -. [. A / x ]. ph ) ) }
16		nfv 1843	. . . 4 |- F/ y ( ( x e. { A } /\ [. A / x ]. ph ) \/ ( x e. (/) /\ -. [. A / x ]. ph ) )
17		nfv 1843	. . . . . 6 |- F/ x y e. { A }
18		nfsbc1v 3455	. . . . . 6 |- F/ x [. A / x ]. ph
19	17, 18	nfan 1828	. . . . 5 |- F/ x ( y e. { A } /\ [. A / x ]. ph )
20		nfv 1843	. . . . . 6 |- F/ x y e. (/)
21	18	nfn 1784	. . . . . 6 |- F/ x -. [. A / x ]. ph
22	20, 21	nfan 1828	. . . . 5 |- F/ x ( y e. (/) /\ -. [. A / x ]. ph )
23	19, 22	nfor 1834	. . . 4 |- F/ x ( ( y e. { A } /\ [. A / x ]. ph ) \/ ( y e. (/) /\ -. [. A / x ]. ph ) )
24		eleq1 2689	. . . . . 6 |- ( x = y -> ( x e. { A } <-> y e. { A } ) )
25	24	anbi1d 741	. . . . 5 |- ( x = y -> ( ( x e. { A } /\ [. A / x ]. ph ) <-> ( y e. { A } /\ [. A / x ]. ph ) ) )
26		eleq1 2689	. . . . . 6 |- ( x = y -> ( x e. (/) <-> y e. (/) ) )
27	26	anbi1d 741	. . . . 5 |- ( x = y -> ( ( x e. (/) /\ -. [. A / x ]. ph ) <-> ( y e. (/) /\ -. [. A / x ]. ph ) ) )
28	25, 27	orbi12d 746	. . . 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	cbvab 2746	. . 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 2644	. 2 |- { x | ( x e. { A } /\ ph ) } = { y | ( ( y e. { A } /\ [. A / x ]. ph ) \/ ( y e. (/) /\ -. [. A / x ]. ph ) ) }
31		df-rab 2921	. 2 |- { x e. { A } | ph } = { x | ( x e. { A } /\ ph ) }
32		df-if 4087	. 2 |- if ( [. A / x ]. ph , { A } , (/) ) = { y | ( ( y e. { A } /\ [. A / x ]. ph ) \/ ( y e. (/) /\ -. [. A / x ]. ph ) ) }
33	30, 31, 32	3eqtr4i 2654	1 |- { x e. { A } | ph } = if ( [. A / x ]. ph , { A } , (/) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   {crab 2916   [.wsbc 3435   (/)c0 3915   ifcif 4086   {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-nul 3916  df-if 4087  df-sn 4178

This theorem is referenced by:  rabsnif  4258  rabrsn  4259