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Theorem suppsnop 6831
Description: The support of a singleton of an ordered pair. (Contributed by AV, 12-Apr-2019.)
Hypothesis
Ref Expression
suppsnop.f  |-  F  =  { <. X ,  Y >. }
Assertion
Ref Expression
suppsnop  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  ( F supp  Z )  =  if ( Y  =  Z ,  (/) ,  { X } ) )

Proof of Theorem suppsnop
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 f1osng 5762 . . . . . . 7  |-  ( ( X  e.  V  /\  Y  e.  W )  ->  { <. X ,  Y >. } : { X }
-1-1-onto-> { Y } )
2 f1of 5724 . . . . . . 7  |-  ( {
<. X ,  Y >. } : { X } -1-1-onto-> { Y }  ->  { <. X ,  Y >. } : { X } --> { Y } )
31, 2syl 16 . . . . . 6  |-  ( ( X  e.  V  /\  Y  e.  W )  ->  { <. X ,  Y >. } : { X }
--> { Y } )
433adant3 1014 . . . . 5  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  { <. X ,  Y >. } : { X }
--> { Y } )
5 suppsnop.f . . . . . 6  |-  F  =  { <. X ,  Y >. }
65feq1i 5631 . . . . 5  |-  ( F : { X } --> { Y }  <->  { <. X ,  Y >. } : { X } --> { Y }
)
74, 6sylibr 212 . . . 4  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  F : { X }
--> { Y } )
8 snex 4603 . . . . 5  |-  { X }  e.  _V
98a1i 11 . . . 4  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  { X }  e.  _V )
10 fex 6046 . . . 4  |-  ( ( F : { X }
--> { Y }  /\  { X }  e.  _V )  ->  F  e.  _V )
117, 9, 10syl2anc 659 . . 3  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  F  e.  _V )
12 simp3 996 . . 3  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  Z  e.  U )
13 suppval 6819 . . 3  |-  ( ( F  e.  _V  /\  Z  e.  U )  ->  ( F supp  Z )  =  { x  e. 
dom  F  |  ( F " { x }
)  =/=  { Z } } )
1411, 12, 13syl2anc 659 . 2  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  ( F supp  Z )  =  { x  e. 
dom  F  |  ( F " { x }
)  =/=  { Z } } )
155a1i 11 . . . . . 6  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  F  =  { <. X ,  Y >. } )
1615dmeqd 5118 . . . . 5  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  dom  F  =  dom  {
<. X ,  Y >. } )
17 dmsnopg 5387 . . . . . 6  |-  ( Y  e.  W  ->  dom  {
<. X ,  Y >. }  =  { X }
)
18173ad2ant2 1016 . . . . 5  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  dom  { <. X ,  Y >. }  =  { X } )
1916, 18eqtrd 2423 . . . 4  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  dom  F  =  { X } )
20 rabeq 3028 . . . 4  |-  ( dom 
F  =  { X }  ->  { x  e. 
dom  F  |  ( F " { x }
)  =/=  { Z } }  =  {
x  e.  { X }  |  ( F " { x } )  =/=  { Z } } )
2119, 20syl 16 . . 3  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  { x  e.  dom  F  |  ( F " { x } )  =/=  { Z } }  =  { x  e.  { X }  | 
( F " {
x } )  =/= 
{ Z } }
)
22 sneq 3954 . . . . . 6  |-  ( x  =  X  ->  { x }  =  { X } )
2322imaeq2d 5249 . . . . 5  |-  ( x  =  X  ->  ( F " { x }
)  =  ( F
" { X }
) )
2423neeq1d 2659 . . . 4  |-  ( x  =  X  ->  (
( F " {
x } )  =/= 
{ Z }  <->  ( F " { X } )  =/=  { Z }
) )
2524rabsnif 4013 . . 3  |-  { x  e.  { X }  | 
( F " {
x } )  =/= 
{ Z } }  =  if ( ( F
" { X }
)  =/=  { Z } ,  { X } ,  (/) )
2621, 25syl6eq 2439 . 2  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  { x  e.  dom  F  |  ( F " { x } )  =/=  { Z } }  =  if (
( F " { X } )  =/=  { Z } ,  { X } ,  (/) ) )
27 fnsng 5543 . . . . . . . . 9  |-  ( ( X  e.  V  /\  Y  e.  W )  ->  { <. X ,  Y >. }  Fn  { X } )
28273adant3 1014 . . . . . . . 8  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  { <. X ,  Y >. }  Fn  { X } )
295eqcomi 2395 . . . . . . . . . 10  |-  { <. X ,  Y >. }  =  F
3029a1i 11 . . . . . . . . 9  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  { <. X ,  Y >. }  =  F )
3130fneq1d 5579 . . . . . . . 8  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  ( { <. X ,  Y >. }  Fn  { X }  <->  F  Fn  { X } ) )
3228, 31mpbid 210 . . . . . . 7  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  F  Fn  { X } )
33 snidg 3970 . . . . . . . 8  |-  ( X  e.  V  ->  X  e.  { X } )
34333ad2ant1 1015 . . . . . . 7  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  X  e.  { X } )
35 fnsnfv 5834 . . . . . . . 8  |-  ( ( F  Fn  { X }  /\  X  e.  { X } )  ->  { ( F `  X ) }  =  ( F
" { X }
) )
3635eqcomd 2390 . . . . . . 7  |-  ( ( F  Fn  { X }  /\  X  e.  { X } )  ->  ( F " { X }
)  =  { ( F `  X ) } )
3732, 34, 36syl2anc 659 . . . . . 6  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  ( F " { X } )  =  {
( F `  X
) } )
3837neeq1d 2659 . . . . 5  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  ( ( F " { X } )  =/= 
{ Z }  <->  { ( F `  X ) }  =/=  { Z }
) )
395fveq1i 5775 . . . . . . . 8  |-  ( F `
 X )  =  ( { <. X ,  Y >. } `  X
)
40 fvsng 6007 . . . . . . . . 9  |-  ( ( X  e.  V  /\  Y  e.  W )  ->  ( { <. X ,  Y >. } `  X
)  =  Y )
41403adant3 1014 . . . . . . . 8  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  ( { <. X ,  Y >. } `  X
)  =  Y )
4239, 41syl5eq 2435 . . . . . . 7  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  ( F `  X
)  =  Y )
4342sneqd 3956 . . . . . 6  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  { ( F `  X ) }  =  { Y } )
4443neeq1d 2659 . . . . 5  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  ( { ( F `
 X ) }  =/=  { Z }  <->  { Y }  =/=  { Z } ) )
45 sneqbg 4114 . . . . . . 7  |-  ( Y  e.  W  ->  ( { Y }  =  { Z }  <->  Y  =  Z
) )
46453ad2ant2 1016 . . . . . 6  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  ( { Y }  =  { Z }  <->  Y  =  Z ) )
4746necon3abid 2628 . . . . 5  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  ( { Y }  =/=  { Z }  <->  -.  Y  =  Z ) )
4838, 44, 473bitrd 279 . . . 4  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  ( ( F " { X } )  =/= 
{ Z }  <->  -.  Y  =  Z ) )
4948ifbid 3879 . . 3  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  if ( ( F
" { X }
)  =/=  { Z } ,  { X } ,  (/) )  =  if ( -.  Y  =  Z ,  { X } ,  (/) ) )
50 ifnot 3902 . . 3  |-  if ( -.  Y  =  Z ,  { X } ,  (/) )  =  if ( Y  =  Z ,  (/) ,  { X } )
5149, 50syl6eq 2439 . 2  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  if ( ( F
" { X }
)  =/=  { Z } ,  { X } ,  (/) )  =  if ( Y  =  Z ,  (/) ,  { X } ) )
5214, 26, 513eqtrd 2427 1  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  ( F supp  Z )  =  if ( Y  =  Z ,  (/) ,  { X } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826    =/= wne 2577   {crab 2736   _Vcvv 3034   (/)c0 3711   ifcif 3857   {csn 3944   <.cop 3950   dom cdm 4913   "cima 4916    Fn wfn 5491   -->wf 5492   -1-1-onto->wf1o 5495   ` cfv 5496  (class class class)co 6196   supp csupp 6817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-supp 6818
This theorem is referenced by: (None)
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