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Theorem rabsnifsb 4031
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.
StepHypRef Expression
1 elsni 3985 . . . . . . . 8  |-  ( x  e.  { A }  ->  x  =  A )
2 sbceq1a 3266 . . . . . . . . 9  |-  ( x  =  A  ->  ( ph 
<-> 
[. A  /  x ]. ph ) )
32biimpd 212 . . . . . . . 8  |-  ( x  =  A  ->  ( ph  ->  [. A  /  x ]. ph ) )
41, 3syl 17 . . . . . . 7  |-  ( x  e.  { A }  ->  ( ph  ->  [. A  /  x ]. ph )
)
54imdistani 704 . . . . . 6  |-  ( ( x  e.  { A }  /\  ph )  -> 
( x  e.  { A }  /\  [. A  /  x ]. ph )
)
65orcd 399 . . . . 5  |-  ( ( x  e.  { A }  /\  ph )  -> 
( ( x  e. 
{ A }  /\  [. A  /  x ]. ph )  \/  ( x  e.  (/)  /\  -.  [. A  /  x ]. ph )
) )
72bicomd 206 . . . . . . . . 9  |-  ( x  =  A  ->  ( [. A  /  x ]. ph  <->  ph ) )
81, 7syl 17 . . . . . . . 8  |-  ( x  e.  { A }  ->  ( [. A  /  x ]. ph  <->  ph ) )
98biimpd 212 . . . . . . 7  |-  ( x  e.  { A }  ->  ( [. A  /  x ]. ph  ->  ph )
)
109imdistani 704 . . . . . 6  |-  ( ( x  e.  { A }  /\  [. A  /  x ]. ph )  -> 
( x  e.  { A }  /\  ph )
)
11 noel 3726 . . . . . . . 8  |-  -.  x  e.  (/)
1211pm2.21i 136 . . . . . . 7  |-  ( x  e.  (/)  ->  ( x  e.  { A }  /\  ph ) )
1312adantr 472 . . . . . 6  |-  ( ( x  e.  (/)  /\  -.  [. A  /  x ]. ph )  ->  ( x  e.  { A }  /\  ph ) )
1410, 13jaoi 386 . . . . 5  |-  ( ( ( x  e.  { A }  /\  [. A  /  x ]. ph )  \/  ( x  e.  (/)  /\ 
-.  [. A  /  x ]. ph ) )  -> 
( x  e.  { A }  /\  ph )
)
156, 14impbii 192 . . . 4  |-  ( ( x  e.  { A }  /\  ph )  <->  ( (
x  e.  { A }  /\  [. A  /  x ]. ph )  \/  ( x  e.  (/)  /\ 
-.  [. A  /  x ]. ph ) ) )
1615abbii 2587 . . 3  |-  { x  |  ( x  e. 
{ A }  /\  ph ) }  =  {
x  |  ( ( x  e.  { A }  /\  [. A  /  x ]. ph )  \/  ( x  e.  (/)  /\ 
-.  [. A  /  x ]. ph ) ) }
17 nfv 1769 . . . 4  |-  F/ y ( ( x  e. 
{ A }  /\  [. A  /  x ]. ph )  \/  ( x  e.  (/)  /\  -.  [. A  /  x ]. ph )
)
18 nfv 1769 . . . . . 6  |-  F/ x  y  e.  { A }
19 nfsbc1v 3275 . . . . . 6  |-  F/ x [. A  /  x ]. ph
2018, 19nfan 2031 . . . . 5  |-  F/ x
( y  e.  { A }  /\  [. A  /  x ]. ph )
21 nfv 1769 . . . . . 6  |-  F/ x  y  e.  (/)
2219nfn 2003 . . . . . 6  |-  F/ x  -.  [. A  /  x ]. ph
2321, 22nfan 2031 . . . . 5  |-  F/ x
( y  e.  (/)  /\ 
-.  [. A  /  x ]. ph )
2420, 23nfor 2038 . . . 4  |-  F/ x
( ( y  e. 
{ A }  /\  [. A  /  x ]. ph )  \/  ( y  e.  (/)  /\  -.  [. A  /  x ]. ph )
)
25 eleq1 2537 . . . . . 6  |-  ( x  =  y  ->  (
x  e.  { A } 
<->  y  e.  { A } ) )
2625anbi1d 719 . . . . 5  |-  ( x  =  y  ->  (
( x  e.  { A }  /\  [. A  /  x ]. ph )  <->  ( y  e.  { A }  /\  [. A  /  x ]. ph ) ) )
27 eleq1 2537 . . . . . 6  |-  ( x  =  y  ->  (
x  e.  (/)  <->  y  e.  (/) ) )
2827anbi1d 719 . . . . 5  |-  ( x  =  y  ->  (
( x  e.  (/)  /\ 
-.  [. A  /  x ]. ph )  <->  ( y  e.  (/)  /\  -.  [. A  /  x ]. ph )
) )
2926, 28orbi12d 724 . . . 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 ) ) ) )
3017, 24, 29cbvab 2594 . . 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 ) ) }
3116, 30eqtri 2493 . 2  |-  { x  |  ( x  e. 
{ A }  /\  ph ) }  =  {
y  |  ( ( y  e.  { A }  /\  [. A  /  x ]. ph )  \/  ( y  e.  (/)  /\ 
-.  [. A  /  x ]. ph ) ) }
32 df-rab 2765 . 2  |-  { x  e.  { A }  |  ph }  =  { x  |  ( x  e. 
{ A }  /\  ph ) }
33 df-if 3873 . 2  |-  if (
[. A  /  x ]. ph ,  { A } ,  (/) )  =  { y  |  ( ( y  e.  { A }  /\  [. A  /  x ]. ph )  \/  ( y  e.  (/)  /\ 
-.  [. A  /  x ]. ph ) ) }
3431, 32, 333eqtr4i 2503 1  |-  { x  e.  { A }  |  ph }  =  if (
[. A  /  x ]. ph ,  { A } ,  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    = wceq 1452    e. wcel 1904   {cab 2457   {crab 2760   [.wsbc 3255   (/)c0 3722   ifcif 3872   {csn 3959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-nul 3723  df-if 3873  df-sn 3960
This theorem is referenced by:  rabsnif  4032  rabrsn  4033
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