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Theorem suppsnop 6917
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 5844 . . . . . . 7  |-  ( ( X  e.  V  /\  Y  e.  W )  ->  { <. X ,  Y >. } : { X }
-1-1-onto-> { Y } )
2 f1of 5806 . . . . . . 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 1017 . . . . 5  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  { <. X ,  Y >. } : { X }
--> { Y } )
5 suppsnop.f . . . . . 6  |-  F  =  { <. X ,  Y >. }
65feq1i 5713 . . . . 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 4678 . . . . 5  |-  { X }  e.  _V
98a1i 11 . . . 4  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  { X }  e.  _V )
10 fex 6130 . . . 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 999 . . 3  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  Z  e.  U )
13 suppval 6905 . . 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 . . . . . 6  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  F  =  { <. X ,  Y >. } )
1615dmeqd 5195 . . . . 5  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  dom  F  =  dom  {
<. X ,  Y >. } )
17 dmsnopg 5469 . . . . . 6  |-  ( Y  e.  W  ->  dom  {
<. X ,  Y >. }  =  { X }
)
18173ad2ant2 1019 . . . . 5  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  dom  { <. X ,  Y >. }  =  { X } )
1916, 18eqtrd 2484 . . . 4  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  dom  F  =  { X } )
20 rabeq 3089 . . . 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 4024 . . . . . 6  |-  ( x  =  X  ->  { x }  =  { X } )
2322imaeq2d 5327 . . . . 5  |-  ( x  =  X  ->  ( F " { x }
)  =  ( F
" { X }
) )
2423neeq1d 2720 . . . 4  |-  ( x  =  X  ->  (
( F " {
x } )  =/= 
{ Z }  <->  ( F " { X } )  =/=  { Z }
) )
2524rabsnif 4084 . . 3  |-  { x  e.  { X }  | 
( F " {
x } )  =/= 
{ Z } }  =  if ( ( F
" { X }
)  =/=  { Z } ,  { X } ,  (/) )
2621, 25syl6eq 2500 . 2  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  { x  e.  dom  F  |  ( F " { x } )  =/=  { Z } }  =  if (
( F " { X } )  =/=  { Z } ,  { X } ,  (/) ) )
27 fnsng 5625 . . . . . . . . 9  |-  ( ( X  e.  V  /\  Y  e.  W )  ->  { <. X ,  Y >. }  Fn  { X } )
28273adant3 1017 . . . . . . . 8  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  { <. X ,  Y >. }  Fn  { X } )
295eqcomi 2456 . . . . . . . . . 10  |-  { <. X ,  Y >. }  =  F
3029a1i 11 . . . . . . . . 9  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  { <. X ,  Y >. }  =  F )
3130fneq1d 5661 . . . . . . . 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 4040 . . . . . . . 8  |-  ( X  e.  V  ->  X  e.  { X } )
34333ad2ant1 1018 . . . . . . 7  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  X  e.  { X } )
35 fnsnfv 5918 . . . . . . . 8  |-  ( ( F  Fn  { X }  /\  X  e.  { X } )  ->  { ( F `  X ) }  =  ( F
" { X }
) )
3635eqcomd 2451 . . . . . . 7  |-  ( ( F  Fn  { X }  /\  X  e.  { X } )  ->  ( F " { X }
)  =  { ( F `  X ) } )
3732, 34, 36syl2anc 661 . . . . . 6  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  ( F " { X } )  =  {
( F `  X
) } )
3837neeq1d 2720 . . . . 5  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  ( ( F " { X } )  =/= 
{ Z }  <->  { ( F `  X ) }  =/=  { Z }
) )
395fveq1i 5857 . . . . . . . 8  |-  ( F `
 X )  =  ( { <. X ,  Y >. } `  X
)
40 fvsng 6090 . . . . . . . . 9  |-  ( ( X  e.  V  /\  Y  e.  W )  ->  ( { <. X ,  Y >. } `  X
)  =  Y )
41403adant3 1017 . . . . . . . 8  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  ( { <. X ,  Y >. } `  X
)  =  Y )
4239, 41syl5eq 2496 . . . . . . 7  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  ( F `  X
)  =  Y )
4342sneqd 4026 . . . . . 6  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  { ( F `  X ) }  =  { Y } )
4443neeq1d 2720 . . . . 5  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  ( { ( F `
 X ) }  =/=  { Z }  <->  { Y }  =/=  { Z } ) )
45 sneqbg 4185 . . . . . . 7  |-  ( Y  e.  W  ->  ( { Y }  =  { Z }  <->  Y  =  Z
) )
46453ad2ant2 1019 . . . . . 6  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  ( { Y }  =  { Z }  <->  Y  =  Z ) )
4746necon3abid 2689 . . . . 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 3948 . . 3  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  if ( ( F
" { X }
)  =/=  { Z } ,  { X } ,  (/) )  =  if ( -.  Y  =  Z ,  { X } ,  (/) ) )
50 ifnot 3971 . . 3  |-  if ( -.  Y  =  Z ,  { X } ,  (/) )  =  if ( Y  =  Z ,  (/) ,  { X } )
5149, 50syl6eq 2500 . 2  |-  ( ( X  e.  V  /\  Y  e.  W  /\  Z  e.  U )  ->  if ( ( F
" { X }
)  =/=  { Z } ,  { X } ,  (/) )  =  if ( Y  =  Z ,  (/) ,  { X } ) )
5214, 26, 513eqtrd 2488 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 974    = wceq 1383    e. wcel 1804    =/= wne 2638   {crab 2797   _Vcvv 3095   (/)c0 3770   ifcif 3926   {csn 4014   <.cop 4020   dom cdm 4989   "cima 4992    Fn wfn 5573   -->wf 5574   -1-1-onto->wf1o 5577   ` cfv 5578  (class class class)co 6281   supp csupp 6903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-supp 6904
This theorem is referenced by: (None)
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