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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.
StepHypRef Expression
1 elsni 4052 . . . . . . . 8  |-  ( x  e.  { A }  ->  x  =  A )
2 sbceq1a 3342 . . . . . . . . 9  |-  ( x  =  A  ->  ( ph 
<-> 
[. A  /  x ]. ph ) )
32biimpd 207 . . . . . . . 8  |-  ( x  =  A  ->  ( ph  ->  [. A  /  x ]. ph ) )
41, 3syl 16 . . . . . . 7  |-  ( x  e.  { A }  ->  ( ph  ->  [. A  /  x ]. ph )
)
54imdistani 690 . . . . . 6  |-  ( ( x  e.  { A }  /\  ph )  -> 
( x  e.  { A }  /\  [. A  /  x ]. ph )
)
65orcd 392 . . . . 5  |-  ( ( x  e.  { A }  /\  ph )  -> 
( ( x  e. 
{ A }  /\  [. A  /  x ]. ph )  \/  ( x  e.  (/)  /\  -.  [. A  /  x ]. ph )
) )
72bicomd 201 . . . . . . . . 9  |-  ( x  =  A  ->  ( [. A  /  x ]. ph  <->  ph ) )
81, 7syl 16 . . . . . . . 8  |-  ( x  e.  { A }  ->  ( [. A  /  x ]. ph  <->  ph ) )
98biimpd 207 . . . . . . 7  |-  ( x  e.  { A }  ->  ( [. A  /  x ]. ph  ->  ph )
)
109imdistani 690 . . . . . 6  |-  ( ( x  e.  { A }  /\  [. A  /  x ]. ph )  -> 
( x  e.  { A }  /\  ph )
)
11 noel 3789 . . . . . . . 8  |-  -.  x  e.  (/)
1211pm2.21i 131 . . . . . . 7  |-  ( x  e.  (/)  ->  ( x  e.  { A }  /\  ph ) )
1312adantr 465 . . . . . 6  |-  ( ( x  e.  (/)  /\  -.  [. A  /  x ]. ph )  ->  ( x  e.  { A }  /\  ph ) )
1410, 13jaoi 379 . . . . 5  |-  ( ( ( x  e.  { A }  /\  [. A  /  x ]. ph )  \/  ( x  e.  (/)  /\ 
-.  [. A  /  x ]. ph ) )  -> 
( x  e.  { A }  /\  ph )
)
156, 14impbii 188 . . . 4  |-  ( ( x  e.  { A }  /\  ph )  <->  ( (
x  e.  { A }  /\  [. A  /  x ]. ph )  \/  ( x  e.  (/)  /\ 
-.  [. A  /  x ]. ph ) ) )
1615abbii 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
2018, 19nfan 1875 . . . . 5  |-  F/ x
( y  e.  { A }  /\  [. A  /  x ]. ph )
21 nfv 1683 . . . . . 6  |-  F/ x  y  e.  (/)
2219nfn 1849 . . . . . 6  |-  F/ x  -.  [. A  /  x ]. ph
2321, 22nfan 1875 . . . . 5  |-  F/ x
( y  e.  (/)  /\ 
-.  [. A  /  x ]. ph )
2420, 23nfor 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 } ) )
2625anbi1d 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.  (/) ) )
2827anbi1d 704 . . . . 5  |-  ( x  =  y  ->  (
( x  e.  (/)  /\ 
-.  [. A  /  x ]. ph )  <->  ( y  e.  (/)  /\  -.  [. A  /  x ]. ph )
) )
2926, 28orbi12d 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 ) ) ) )
3017, 24, 29cbvab 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 ) ) }
3116, 30eqtri 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 ) ) }
3431, 32, 333eqtr4i 2506 1  |-  { x  e.  { A }  |  ph }  =  if (
[. A  /  x ]. ph ,  { A } ,  (/) )
Colors of variables: wff setvar class
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
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