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Theorem suppimacnv 6696
Description: Support sets of functions expressed by inverse images. (Contributed by AV, 31-Mar-2019.) (Revised by AV, 7-Apr-2019.)
Assertion
Ref Expression
suppimacnv  |-  ( ( R  e.  V  /\  Z  e.  W )  ->  ( R supp  Z )  =  ( `' R " ( _V  \  { Z } ) ) )

Proof of Theorem suppimacnv
Dummy variables  x  y  s  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq2 4291 . . . . . . . 8  |-  ( t  =  s  ->  (
x R t  <->  x R
s ) )
21cbvexvw 1748 . . . . . . 7  |-  ( E. t  x R t  <->  E. s  x R
s )
3 breq2 4291 . . . . . . . . . . . . . 14  |-  ( s  =  Z  ->  (
x R s  <->  x R Z ) )
43anbi1d 704 . . . . . . . . . . . . 13  |-  ( s  =  Z  ->  (
( x R s  /\  ( x R t  <->  t  =/=  Z
) )  <->  ( x R Z  /\  (
x R t  <->  t  =/=  Z ) ) ) )
5 bianir 958 . . . . . . . . . . . . . . . . . 18  |-  ( ( t  =/=  Z  /\  ( x R t  <-> 
t  =/=  Z ) )  ->  x R
t )
6 vex 2970 . . . . . . . . . . . . . . . . . . . 20  |-  t  e. 
_V
7 breq2 4291 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  =  t  ->  (
x R y  <->  x R
t ) )
8 neeq1 2611 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  =  t  ->  (
y  =/=  Z  <->  t  =/=  Z ) )
97, 8anbi12d 710 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  =  t  ->  (
( x R y  /\  y  =/=  Z
)  <->  ( x R t  /\  t  =/= 
Z ) ) )
106, 9spcev 3059 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x R t  /\  t  =/=  Z )  ->  E. y ( x R y  /\  y  =/= 
Z ) )
1110ex 434 . . . . . . . . . . . . . . . . . 18  |-  ( x R t  ->  (
t  =/=  Z  ->  E. y ( x R y  /\  y  =/= 
Z ) ) )
125, 11syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( t  =/=  Z  /\  ( x R t  <-> 
t  =/=  Z ) )  ->  ( t  =/=  Z  ->  E. y
( x R y  /\  y  =/=  Z
) ) )
1312ex 434 . . . . . . . . . . . . . . . 16  |-  ( t  =/=  Z  ->  (
( x R t  <-> 
t  =/=  Z )  ->  ( t  =/= 
Z  ->  E. y
( x R y  /\  y  =/=  Z
) ) ) )
1413pm2.43a 49 . . . . . . . . . . . . . . 15  |-  ( t  =/=  Z  ->  (
( x R t  <-> 
t  =/=  Z )  ->  E. y ( x R y  /\  y  =/=  Z ) ) )
1514adantld 467 . . . . . . . . . . . . . 14  |-  ( t  =/=  Z  ->  (
( x R Z  /\  ( x R t  <->  t  =/=  Z
) )  ->  E. y
( x R y  /\  y  =/=  Z
) ) )
16 nne 2607 . . . . . . . . . . . . . . . 16  |-  ( -.  t  =/=  Z  <->  t  =  Z )
17 notbi 295 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x R t  <->  t  =/=  Z )  <->  ( -.  x R t  <->  -.  t  =/=  Z ) )
18 bianir 958 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( -.  t  =/=  Z  /\  ( -.  x R t  <->  -.  t  =/=  Z ) )  ->  -.  x R t )
19 breq2 4291 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( Z  =  t  ->  (
x R Z  <->  x R
t ) )
2019eqcoms 2441 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( t  =  Z  ->  (
x R Z  <->  x R
t ) )
21 pm2.24 109 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( x R t  ->  ( -.  x R t  ->  E. y ( x R y  /\  y  =/= 
Z ) ) )
2220, 21syl6bi 228 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( t  =  Z  ->  (
x R Z  -> 
( -.  x R t  ->  E. y
( x R y  /\  y  =/=  Z
) ) ) )
2322com13 80 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( -.  x R t  -> 
( x R Z  ->  ( t  =  Z  ->  E. y
( x R y  /\  y  =/=  Z
) ) ) )
2418, 23syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( -.  t  =/=  Z  /\  ( -.  x R t  <->  -.  t  =/=  Z ) )  ->  (
x R Z  -> 
( t  =  Z  ->  E. y ( x R y  /\  y  =/=  Z ) ) ) )
2524ex 434 . . . . . . . . . . . . . . . . . . . 20  |-  ( -.  t  =/=  Z  -> 
( ( -.  x R t  <->  -.  t  =/=  Z )  ->  (
x R Z  -> 
( t  =  Z  ->  E. y ( x R y  /\  y  =/=  Z ) ) ) ) )
2617, 25syl5bi 217 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  t  =/=  Z  -> 
( ( x R t  <->  t  =/=  Z
)  ->  ( x R Z  ->  ( t  =  Z  ->  E. y
( x R y  /\  y  =/=  Z
) ) ) ) )
2726com13 80 . . . . . . . . . . . . . . . . . 18  |-  ( x R Z  ->  (
( x R t  <-> 
t  =/=  Z )  ->  ( -.  t  =/=  Z  ->  ( t  =  Z  ->  E. y
( x R y  /\  y  =/=  Z
) ) ) ) )
2827imp 429 . . . . . . . . . . . . . . . . 17  |-  ( ( x R Z  /\  ( x R t  <-> 
t  =/=  Z ) )  ->  ( -.  t  =/=  Z  ->  (
t  =  Z  ->  E. y ( x R y  /\  y  =/= 
Z ) ) ) )
2928com13 80 . . . . . . . . . . . . . . . 16  |-  ( t  =  Z  ->  ( -.  t  =/=  Z  ->  ( ( x R Z  /\  ( x R t  <->  t  =/=  Z ) )  ->  E. y
( x R y  /\  y  =/=  Z
) ) ) )
3016, 29sylbi 195 . . . . . . . . . . . . . . 15  |-  ( -.  t  =/=  Z  -> 
( -.  t  =/= 
Z  ->  ( (
x R Z  /\  ( x R t  <-> 
t  =/=  Z ) )  ->  E. y
( x R y  /\  y  =/=  Z
) ) ) )
3130pm2.43i 47 . . . . . . . . . . . . . 14  |-  ( -.  t  =/=  Z  -> 
( ( x R Z  /\  ( x R t  <->  t  =/=  Z ) )  ->  E. y
( x R y  /\  y  =/=  Z
) ) )
3215, 31pm2.61i 164 . . . . . . . . . . . . 13  |-  ( ( x R Z  /\  ( x R t  <-> 
t  =/=  Z ) )  ->  E. y
( x R y  /\  y  =/=  Z
) )
334, 32syl6bi 228 . . . . . . . . . . . 12  |-  ( s  =  Z  ->  (
( x R s  /\  ( x R t  <->  t  =/=  Z
) )  ->  E. y
( x R y  /\  y  =/=  Z
) ) )
34 vex 2970 . . . . . . . . . . . . . . . 16  |-  s  e. 
_V
35 breq2 4291 . . . . . . . . . . . . . . . . 17  |-  ( y  =  s  ->  (
x R y  <->  x R
s ) )
36 neeq1 2611 . . . . . . . . . . . . . . . . 17  |-  ( y  =  s  ->  (
y  =/=  Z  <->  s  =/=  Z ) )
3735, 36anbi12d 710 . . . . . . . . . . . . . . . 16  |-  ( y  =  s  ->  (
( x R y  /\  y  =/=  Z
)  <->  ( x R s  /\  s  =/= 
Z ) ) )
3834, 37spcev 3059 . . . . . . . . . . . . . . 15  |-  ( ( x R s  /\  s  =/=  Z )  ->  E. y ( x R y  /\  y  =/= 
Z ) )
3938ex 434 . . . . . . . . . . . . . 14  |-  ( x R s  ->  (
s  =/=  Z  ->  E. y ( x R y  /\  y  =/= 
Z ) ) )
4039adantr 465 . . . . . . . . . . . . 13  |-  ( ( x R s  /\  ( x R t  <-> 
t  =/=  Z ) )  ->  ( s  =/=  Z  ->  E. y
( x R y  /\  y  =/=  Z
) ) )
4140com12 31 . . . . . . . . . . . 12  |-  ( s  =/=  Z  ->  (
( x R s  /\  ( x R t  <->  t  =/=  Z
) )  ->  E. y
( x R y  /\  y  =/=  Z
) ) )
4233, 41pm2.61ine 2682 . . . . . . . . . . 11  |-  ( ( x R s  /\  ( x R t  <-> 
t  =/=  Z ) )  ->  E. y
( x R y  /\  y  =/=  Z
) )
4342expcom 435 . . . . . . . . . 10  |-  ( ( x R t  <->  t  =/=  Z )  ->  ( x R s  ->  E. y
( x R y  /\  y  =/=  Z
) ) )
4443exlimiv 1688 . . . . . . . . 9  |-  ( E. t ( x R t  <->  t  =/=  Z
)  ->  ( x R s  ->  E. y
( x R y  /\  y  =/=  Z
) ) )
4544com12 31 . . . . . . . 8  |-  ( x R s  ->  ( E. t ( x R t  <->  t  =/=  Z
)  ->  E. y
( x R y  /\  y  =/=  Z
) ) )
4645exlimiv 1688 . . . . . . 7  |-  ( E. s  x R s  ->  ( E. t
( x R t  <-> 
t  =/=  Z )  ->  E. y ( x R y  /\  y  =/=  Z ) ) )
472, 46sylbi 195 . . . . . 6  |-  ( E. t  x R t  ->  ( E. t
( x R t  <-> 
t  =/=  Z )  ->  E. y ( x R y  /\  y  =/=  Z ) ) )
4847imp 429 . . . . 5  |-  ( ( E. t  x R t  /\  E. t
( x R t  <-> 
t  =/=  Z ) )  ->  E. y
( x R y  /\  y  =/=  Z
) )
4948a1i 11 . . . 4  |-  ( ( R  e.  V  /\  Z  e.  W )  ->  ( ( E. t  x R t  /\  E. t ( x R t  <->  t  =/=  Z
) )  ->  E. y
( x R y  /\  y  =/=  Z
) ) )
5049ss2abdv 3420 . . 3  |-  ( ( R  e.  V  /\  Z  e.  W )  ->  { x  |  ( E. t  x R t  /\  E. t
( x R t  <-> 
t  =/=  Z ) ) }  C_  { x  |  E. y ( x R y  /\  y  =/=  Z ) } )
51 suppvalbr 6689 . . 3  |-  ( ( R  e.  V  /\  Z  e.  W )  ->  ( R supp  Z )  =  { x  |  ( E. t  x R t  /\  E. t ( x R t  <->  t  =/=  Z
) ) } )
52 cnvimadfsn 6694 . . . 4  |-  ( `' R " ( _V 
\  { Z }
) )  =  {
x  |  E. y
( x R y  /\  y  =/=  Z
) }
5352a1i 11 . . 3  |-  ( ( R  e.  V  /\  Z  e.  W )  ->  ( `' R "
( _V  \  { Z } ) )  =  { x  |  E. y ( x R y  /\  y  =/= 
Z ) } )
5450, 51, 533sstr4d 3394 . 2  |-  ( ( R  e.  V  /\  Z  e.  W )  ->  ( R supp  Z ) 
C_  ( `' R " ( _V  \  { Z } ) ) )
55 suppimacnvss 6695 . 2  |-  ( ( R  e.  V  /\  Z  e.  W )  ->  ( `' R "
( _V  \  { Z } ) )  C_  ( R supp  Z )
)
5654, 55eqssd 3368 1  |-  ( ( R  e.  V  /\  Z  e.  W )  ->  ( R supp  Z )  =  ( `' R " ( _V  \  { Z } ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756   {cab 2424    =/= wne 2601   _Vcvv 2967    \ cdif 3320   {csn 3872   class class class wbr 4287   `'ccnv 4834   "cima 4838  (class class class)co 6086   supp csupp 6685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-supp 6686
This theorem is referenced by:  frnsuppeq  6697  suppun  6704  mptsuppdifd  6706  supp0cosupp0  6723  imacosupp  6724  fdmfisuppfi  7621  fsuppun  7631  fsuppco  7643  cantnffvalOLD  7863  gsumval3a  16370  gsumzf1o  16382  gsumzaddlem  16399  gsummptfsaddOLD  16406  gsumzmhm  16420  gsumzoppg  16430  dprdvalOLD  16475  mplvalOLD  17479  mdegfvalOLD  21507  deg1val  21542  suppss3  25978  ffsrn  25980  fpwrelmapffslem  25983  eulerpartlemmf  26710  eulerpartlemgf  26714
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