MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  suppsnop Structured version   Unicode version

Theorem suppsnop 6817
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 5790 . . . . . . 7  |-  ( ( X  e.  V  /\  Y  e.  W )  ->  { <. X ,  Y >. } : { X }
-1-1-onto-> { Y } )
2 f1of 5752 . . . . . . 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 1008 . . . . 5  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  { <. X ,  Y >. } : { X }
--> { Y } )
5 suppsnop.f . . . . . 6  |-  F  =  { <. X ,  Y >. }
65feq1i 5662 . . . . 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 4644 . . . . 5  |-  { X }  e.  _V
98a1i 11 . . . 4  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  { X }  e.  _V )
10 fex 6062 . . . 4  |-  ( ( F : { X }
--> { Y }  /\  { X }  e.  _V )  ->  F  e.  _V )
117, 9, 10syl2anc 661 . . 3  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  F  e.  _V )
12 simp3 990 . . 3  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  Z  e.  U )
13 suppval 6805 . . 3  |-  ( ( F  e.  _V  /\  Z  e.  U )  ->  ( F supp  Z )  =  { x  e. 
dom  F  |  ( F " { x }
)  =/=  { Z } } )
1411, 12, 13syl2anc 661 . 2  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  ( F supp  Z )  =  { x  e. 
dom  F  |  ( F " { x }
)  =/=  { Z } } )
155a1i 11 . . . . . . 7  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  F  =  { <. X ,  Y >. } )
1615dmeqd 5153 . . . . . 6  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  dom  F  =  dom  {
<. X ,  Y >. } )
17 dmsnopg 5421 . . . . . . 7  |-  ( Y  e.  W  ->  dom  {
<. X ,  Y >. }  =  { X }
)
18173ad2ant2 1010 . . . . . 6  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  dom  { <. X ,  Y >. }  =  { X } )
1916, 18eqtrd 2495 . . . . 5  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  dom  F  =  { X } )
20 rabeq 3072 . . . . 5  |-  ( dom 
F  =  { X }  ->  { x  e. 
dom  F  |  ( F " { x }
)  =/=  { Z } }  =  {
x  e.  { X }  |  ( F " { x } )  =/=  { Z } } )
2119, 20syl 16 . . . 4  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  { x  e.  dom  F  |  ( F " { x } )  =/=  { Z } }  =  { x  e.  { X }  | 
( F " {
x } )  =/= 
{ Z } }
)
22 sneq 3998 . . . . . . 7  |-  ( x  =  X  ->  { x }  =  { X } )
2322imaeq2d 5280 . . . . . 6  |-  ( x  =  X  ->  ( F " { x }
)  =  ( F
" { X }
) )
2423neeq1d 2729 . . . . 5  |-  ( x  =  X  ->  (
( F " {
x } )  =/= 
{ Z }  <->  ( F " { X } )  =/=  { Z }
) )
2524rabsnif 4055 . . . 4  |-  { x  e.  { X }  | 
( F " {
x } )  =/= 
{ Z } }  =  if ( ( F
" { X }
)  =/=  { Z } ,  { X } ,  (/) )
2621, 25syl6eq 2511 . . 3  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  { x  e.  dom  F  |  ( F " { x } )  =/=  { Z } }  =  if (
( F " { X } )  =/=  { Z } ,  { X } ,  (/) ) )
27 fnsng 5576 . . . . . . . . . 10  |-  ( ( X  e.  V  /\  Y  e.  W )  ->  { <. X ,  Y >. }  Fn  { X } )
28273adant3 1008 . . . . . . . . 9  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  { <. X ,  Y >. }  Fn  { X } )
295eqcomi 2467 . . . . . . . . . . 11  |-  { <. X ,  Y >. }  =  F
3029a1i 11 . . . . . . . . . 10  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  { <. X ,  Y >. }  =  F )
3130fneq1d 5612 . . . . . . . . 9  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  ( { <. X ,  Y >. }  Fn  { X }  <->  F  Fn  { X } ) )
3228, 31mpbid 210 . . . . . . . 8  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  F  Fn  { X } )
33 snidg 4014 . . . . . . . . 9  |-  ( X  e.  V  ->  X  e.  { X } )
34333ad2ant1 1009 . . . . . . . 8  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  X  e.  { X } )
35 fnsnfv 5863 . . . . . . . . 9  |-  ( ( F  Fn  { X }  /\  X  e.  { X } )  ->  { ( F `  X ) }  =  ( F
" { X }
) )
3635eqcomd 2462 . . . . . . . 8  |-  ( ( F  Fn  { X }  /\  X  e.  { X } )  ->  ( F " { X }
)  =  { ( F `  X ) } )
3732, 34, 36syl2anc 661 . . . . . . 7  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  ( F " { X } )  =  {
( F `  X
) } )
3837neeq1d 2729 . . . . . 6  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  ( ( F " { X } )  =/= 
{ Z }  <->  { ( F `  X ) }  =/=  { Z }
) )
395fveq1i 5803 . . . . . . . . 9  |-  ( F `
 X )  =  ( { <. X ,  Y >. } `  X
)
40 fvsng 6024 . . . . . . . . . 10  |-  ( ( X  e.  V  /\  Y  e.  W )  ->  ( { <. X ,  Y >. } `  X
)  =  Y )
41403adant3 1008 . . . . . . . . 9  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  ( { <. X ,  Y >. } `  X
)  =  Y )
4239, 41syl5eq 2507 . . . . . . . 8  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  ( F `  X
)  =  Y )
4342sneqd 4000 . . . . . . 7  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  { ( F `  X ) }  =  { Y } )
4443neeq1d 2729 . . . . . 6  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  ( { ( F `
 X ) }  =/=  { Z }  <->  { Y }  =/=  { Z } ) )
45 sneqbg 4154 . . . . . . . 8  |-  ( Y  e.  W  ->  ( { Y }  =  { Z }  <->  Y  =  Z
) )
46453ad2ant2 1010 . . . . . . 7  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  ( { Y }  =  { Z }  <->  Y  =  Z ) )
4746necon3abid 2698 . . . . . 6  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  ( { Y }  =/=  { Z }  <->  -.  Y  =  Z ) )
4838, 44, 473bitrd 279 . . . . 5  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  ( ( F " { X } )  =/= 
{ Z }  <->  -.  Y  =  Z ) )
4948ifbid 3922 . . . 4  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  if ( ( F
" { X }
)  =/=  { Z } ,  { X } ,  (/) )  =  if ( -.  Y  =  Z ,  { X } ,  (/) ) )
50 ifnot 3945 . . . 4  |-  if ( -.  Y  =  Z ,  { X } ,  (/) )  =  if ( Y  =  Z ,  (/) ,  { X } )
5149, 50syl6eq 2511 . . 3  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  if ( ( F
" { X }
)  =/=  { Z } ,  { X } ,  (/) )  =  if ( Y  =  Z ,  (/) ,  { X } ) )
5226, 51eqtrd 2495 . 2  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  { x  e.  dom  F  |  ( F " { x } )  =/=  { Z } }  =  if ( Y  =  Z ,  (/)
,  { X }
) )
5314, 52eqtrd 2495 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   {crab 2803   _Vcvv 3078   (/)c0 3748   ifcif 3902   {csn 3988   <.cop 3994   dom cdm 4951   "cima 4954    Fn wfn 5524   -->wf 5525   -1-1-onto->wf1o 5528   ` cfv 5529  (class class class)co 6203   supp csupp 6803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-supp 6804
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator