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Theorem suppsnop 6955
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 5876 . . . . . . 7  |-  ( ( X  e.  V  /\  Y  e.  W )  ->  { <. X ,  Y >. } : { X }
-1-1-onto-> { Y } )
2 f1of 5837 . . . . . . 7  |-  ( {
<. X ,  Y >. } : { X } -1-1-onto-> { Y }  ->  { <. X ,  Y >. } : { X } --> { Y } )
31, 2syl 17 . . . . . 6  |-  ( ( X  e.  V  /\  Y  e.  W )  ->  { <. X ,  Y >. } : { X }
--> { Y } )
433adant3 1034 . . . . 5  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  { <. X ,  Y >. } : { X }
--> { Y } )
5 suppsnop.f . . . . . 6  |-  F  =  { <. X ,  Y >. }
65feq1i 5742 . . . . 5  |-  ( F : { X } --> { Y }  <->  { <. X ,  Y >. } : { X } --> { Y }
)
74, 6sylibr 217 . . . 4  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  F : { X }
--> { Y } )
8 snex 4655 . . . . 5  |-  { X }  e.  _V
98a1i 11 . . . 4  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  { X }  e.  _V )
10 fex 6163 . . . 4  |-  ( ( F : { X }
--> { Y }  /\  { X }  e.  _V )  ->  F  e.  _V )
117, 9, 10syl2anc 671 . . 3  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  F  e.  _V )
12 simp3 1016 . . 3  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  Z  e.  U )
13 suppval 6943 . . 3  |-  ( ( F  e.  _V  /\  Z  e.  U )  ->  ( F supp  Z )  =  { x  e. 
dom  F  |  ( F " { x }
)  =/=  { Z } } )
1411, 12, 13syl2anc 671 . 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 5056 . . . . 5  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  dom  F  =  dom  {
<. X ,  Y >. } )
17 dmsnopg 5326 . . . . . 6  |-  ( Y  e.  W  ->  dom  {
<. X ,  Y >. }  =  { X }
)
18173ad2ant2 1036 . . . . 5  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  dom  { <. X ,  Y >. }  =  { X } )
1916, 18eqtrd 2496 . . . 4  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  dom  F  =  { X } )
20 rabeq 3050 . . . 4  |-  ( dom 
F  =  { X }  ->  { x  e. 
dom  F  |  ( F " { x }
)  =/=  { Z } }  =  {
x  e.  { X }  |  ( F " { x } )  =/=  { Z } } )
2119, 20syl 17 . . 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 3990 . . . . . 6  |-  ( x  =  X  ->  { x }  =  { X } )
2322imaeq2d 5187 . . . . 5  |-  ( x  =  X  ->  ( F " { x }
)  =  ( F
" { X }
) )
2423neeq1d 2695 . . . 4  |-  ( x  =  X  ->  (
( F " {
x } )  =/= 
{ Z }  <->  ( F " { X } )  =/=  { Z }
) )
2524rabsnif 4054 . . 3  |-  { x  e.  { X }  | 
( F " {
x } )  =/= 
{ Z } }  =  if ( ( F
" { X }
)  =/=  { Z } ,  { X } ,  (/) )
2621, 25syl6eq 2512 . 2  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  { x  e.  dom  F  |  ( F " { x } )  =/=  { Z } }  =  if (
( F " { X } )  =/=  { Z } ,  { X } ,  (/) ) )
27 fnsng 5648 . . . . . . . . 9  |-  ( ( X  e.  V  /\  Y  e.  W )  ->  { <. X ,  Y >. }  Fn  { X } )
28273adant3 1034 . . . . . . . 8  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  { <. X ,  Y >. }  Fn  { X } )
295eqcomi 2471 . . . . . . . . . 10  |-  { <. X ,  Y >. }  =  F
3029a1i 11 . . . . . . . . 9  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  { <. X ,  Y >. }  =  F )
3130fneq1d 5688 . . . . . . . 8  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  ( { <. X ,  Y >. }  Fn  { X }  <->  F  Fn  { X } ) )
3228, 31mpbid 215 . . . . . . 7  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  F  Fn  { X } )
33 snidg 4006 . . . . . . . 8  |-  ( X  e.  V  ->  X  e.  { X } )
34333ad2ant1 1035 . . . . . . 7  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  X  e.  { X } )
35 fnsnfv 5948 . . . . . . . 8  |-  ( ( F  Fn  { X }  /\  X  e.  { X } )  ->  { ( F `  X ) }  =  ( F
" { X }
) )
3635eqcomd 2468 . . . . . . 7  |-  ( ( F  Fn  { X }  /\  X  e.  { X } )  ->  ( F " { X }
)  =  { ( F `  X ) } )
3732, 34, 36syl2anc 671 . . . . . 6  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  ( F " { X } )  =  {
( F `  X
) } )
3837neeq1d 2695 . . . . 5  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  ( ( F " { X } )  =/= 
{ Z }  <->  { ( F `  X ) }  =/=  { Z }
) )
395fveq1i 5889 . . . . . . . 8  |-  ( F `
 X )  =  ( { <. X ,  Y >. } `  X
)
40 fvsng 6122 . . . . . . . . 9  |-  ( ( X  e.  V  /\  Y  e.  W )  ->  ( { <. X ,  Y >. } `  X
)  =  Y )
41403adant3 1034 . . . . . . . 8  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  ( { <. X ,  Y >. } `  X
)  =  Y )
4239, 41syl5eq 2508 . . . . . . 7  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  ( F `  X
)  =  Y )
4342sneqd 3992 . . . . . 6  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  { ( F `  X ) }  =  { Y } )
4443neeq1d 2695 . . . . 5  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  ( { ( F `
 X ) }  =/=  { Z }  <->  { Y }  =/=  { Z } ) )
45 sneqbg 4155 . . . . . . 7  |-  ( Y  e.  W  ->  ( { Y }  =  { Z }  <->  Y  =  Z
) )
46453ad2ant2 1036 . . . . . 6  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  ( { Y }  =  { Z }  <->  Y  =  Z ) )
4746necon3abid 2672 . . . . 5  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  ( { Y }  =/=  { Z }  <->  -.  Y  =  Z ) )
4838, 44, 473bitrd 287 . . . 4  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  ( ( F " { X } )  =/= 
{ Z }  <->  -.  Y  =  Z ) )
4948ifbid 3915 . . 3  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  if ( ( F
" { X }
)  =/=  { Z } ,  { X } ,  (/) )  =  if ( -.  Y  =  Z ,  { X } ,  (/) ) )
50 ifnot 3938 . . 3  |-  if ( -.  Y  =  Z ,  { X } ,  (/) )  =  if ( Y  =  Z ,  (/) ,  { X } )
5149, 50syl6eq 2512 . 2  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  if ( ( F
" { X }
)  =/=  { Z } ,  { X } ,  (/) )  =  if ( Y  =  Z ,  (/) ,  { X } ) )
5214, 26, 513eqtrd 2500 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 189    /\ wa 375    /\ w3a 991    = wceq 1455    e. wcel 1898    =/= wne 2633   {crab 2753   _Vcvv 3057   (/)c0 3743   ifcif 3893   {csn 3980   <.cop 3986   dom cdm 4853   "cima 4856    Fn wfn 5596   -->wf 5597   -1-1-onto->wf1o 5600   ` cfv 5601  (class class class)co 6315   supp csupp 6941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pr 4653  ax-un 6610
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-supp 6942
This theorem is referenced by: (None)
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