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Theorem suppimacnv 6944
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 4399 . . . . . . . 8  |-  ( t  =  s  ->  (
x R t  <->  x R
s ) )
21cbvexvw 1887 . . . . . . 7  |-  ( E. t  x R t  <->  E. s  x R
s )
3 breq2 4399 . . . . . . . . . . . . . 14  |-  ( s  =  Z  ->  (
x R s  <->  x R Z ) )
43anbi1d 719 . . . . . . . . . . . . 13  |-  ( s  =  Z  ->  (
( x R s  /\  ( x R t  <->  t  =/=  Z
) )  <->  ( x R Z  /\  (
x R t  <->  t  =/=  Z ) ) ) )
5 bianir 978 . . . . . . . . . . . . . . . . . 18  |-  ( ( t  =/=  Z  /\  ( x R t  <-> 
t  =/=  Z ) )  ->  x R
t )
6 vex 3034 . . . . . . . . . . . . . . . . . . . 20  |-  t  e. 
_V
7 breq2 4399 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  =  t  ->  (
x R y  <->  x R
t ) )
8 neeq1 2705 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  =  t  ->  (
y  =/=  Z  <->  t  =/=  Z ) )
97, 8anbi12d 725 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  =  t  ->  (
( x R y  /\  y  =/=  Z
)  <->  ( x R t  /\  t  =/= 
Z ) ) )
106, 9spcev 3127 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x R t  /\  t  =/=  Z )  ->  E. y ( x R y  /\  y  =/= 
Z ) )
1110ex 441 . . . . . . . . . . . . . . . . . 18  |-  ( x R t  ->  (
t  =/=  Z  ->  E. y ( x R y  /\  y  =/= 
Z ) ) )
125, 11syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ( t  =/=  Z  /\  ( x R t  <-> 
t  =/=  Z ) )  ->  ( t  =/=  Z  ->  E. y
( x R y  /\  y  =/=  Z
) ) )
1312ex 441 . . . . . . . . . . . . . . . 16  |-  ( t  =/=  Z  ->  (
( x R t  <-> 
t  =/=  Z )  ->  ( t  =/= 
Z  ->  E. y
( x R y  /\  y  =/=  Z
) ) ) )
1413pm2.43a 50 . . . . . . . . . . . . . . 15  |-  ( t  =/=  Z  ->  (
( x R t  <-> 
t  =/=  Z )  ->  E. y ( x R y  /\  y  =/=  Z ) ) )
1514adantld 474 . . . . . . . . . . . . . 14  |-  ( t  =/=  Z  ->  (
( x R Z  /\  ( x R t  <->  t  =/=  Z
) )  ->  E. y
( x R y  /\  y  =/=  Z
) ) )
16 nne 2647 . . . . . . . . . . . . . . . 16  |-  ( -.  t  =/=  Z  <->  t  =  Z )
17 notbi 302 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x R t  <->  t  =/=  Z )  <->  ( -.  x R t  <->  -.  t  =/=  Z ) )
18 bianir 978 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( -.  t  =/=  Z  /\  ( -.  x R t  <->  -.  t  =/=  Z ) )  ->  -.  x R t )
19 breq2 4399 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( Z  =  t  ->  (
x R Z  <->  x R
t ) )
2019eqcoms 2479 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( t  =  Z  ->  (
x R Z  <->  x R
t ) )
21 pm2.24 112 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( x R t  ->  ( -.  x R t  ->  E. y ( x R y  /\  y  =/= 
Z ) ) )
2220, 21syl6bi 236 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( t  =  Z  ->  (
x R Z  -> 
( -.  x R t  ->  E. y
( x R y  /\  y  =/=  Z
) ) ) )
2322com13 82 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( -.  x R t  -> 
( x R Z  ->  ( t  =  Z  ->  E. y
( x R y  /\  y  =/=  Z
) ) ) )
2418, 23syl 17 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( -.  t  =/=  Z  /\  ( -.  x R t  <->  -.  t  =/=  Z ) )  ->  (
x R Z  -> 
( t  =  Z  ->  E. y ( x R y  /\  y  =/=  Z ) ) ) )
2524ex 441 . . . . . . . . . . . . . . . . . . . 20  |-  ( -.  t  =/=  Z  -> 
( ( -.  x R t  <->  -.  t  =/=  Z )  ->  (
x R Z  -> 
( t  =  Z  ->  E. y ( x R y  /\  y  =/=  Z ) ) ) ) )
2617, 25syl5bi 225 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  t  =/=  Z  -> 
( ( x R t  <->  t  =/=  Z
)  ->  ( x R Z  ->  ( t  =  Z  ->  E. y
( x R y  /\  y  =/=  Z
) ) ) ) )
2726com13 82 . . . . . . . . . . . . . . . . . 18  |-  ( x R Z  ->  (
( x R t  <-> 
t  =/=  Z )  ->  ( -.  t  =/=  Z  ->  ( t  =  Z  ->  E. y
( x R y  /\  y  =/=  Z
) ) ) ) )
2827imp 436 . . . . . . . . . . . . . . . . 17  |-  ( ( x R Z  /\  ( x R t  <-> 
t  =/=  Z ) )  ->  ( -.  t  =/=  Z  ->  (
t  =  Z  ->  E. y ( x R y  /\  y  =/= 
Z ) ) ) )
2928com13 82 . . . . . . . . . . . . . . . 16  |-  ( t  =  Z  ->  ( -.  t  =/=  Z  ->  ( ( x R Z  /\  ( x R t  <->  t  =/=  Z ) )  ->  E. y
( x R y  /\  y  =/=  Z
) ) ) )
3016, 29sylbi 200 . . . . . . . . . . . . . . 15  |-  ( -.  t  =/=  Z  -> 
( -.  t  =/= 
Z  ->  ( (
x R Z  /\  ( x R t  <-> 
t  =/=  Z ) )  ->  E. y
( x R y  /\  y  =/=  Z
) ) ) )
3130pm2.43i 48 . . . . . . . . . . . . . 14  |-  ( -.  t  =/=  Z  -> 
( ( x R Z  /\  ( x R t  <->  t  =/=  Z ) )  ->  E. y
( x R y  /\  y  =/=  Z
) ) )
3215, 31pm2.61i 169 . . . . . . . . . . . . 13  |-  ( ( x R Z  /\  ( x R t  <-> 
t  =/=  Z ) )  ->  E. y
( x R y  /\  y  =/=  Z
) )
334, 32syl6bi 236 . . . . . . . . . . . 12  |-  ( s  =  Z  ->  (
( x R s  /\  ( x R t  <->  t  =/=  Z
) )  ->  E. y
( x R y  /\  y  =/=  Z
) ) )
34 vex 3034 . . . . . . . . . . . . . . . 16  |-  s  e. 
_V
35 breq2 4399 . . . . . . . . . . . . . . . . 17  |-  ( y  =  s  ->  (
x R y  <->  x R
s ) )
36 neeq1 2705 . . . . . . . . . . . . . . . . 17  |-  ( y  =  s  ->  (
y  =/=  Z  <->  s  =/=  Z ) )
3735, 36anbi12d 725 . . . . . . . . . . . . . . . 16  |-  ( y  =  s  ->  (
( x R y  /\  y  =/=  Z
)  <->  ( x R s  /\  s  =/= 
Z ) ) )
3834, 37spcev 3127 . . . . . . . . . . . . . . 15  |-  ( ( x R s  /\  s  =/=  Z )  ->  E. y ( x R y  /\  y  =/= 
Z ) )
3938ex 441 . . . . . . . . . . . . . 14  |-  ( x R s  ->  (
s  =/=  Z  ->  E. y ( x R y  /\  y  =/= 
Z ) ) )
4039adantr 472 . . . . . . . . . . . . 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 2726 . . . . . . . . . . 11  |-  ( ( x R s  /\  ( x R t  <-> 
t  =/=  Z ) )  ->  E. y
( x R y  /\  y  =/=  Z
) )
4342expcom 442 . . . . . . . . . 10  |-  ( ( x R t  <->  t  =/=  Z )  ->  ( x R s  ->  E. y
( x R y  /\  y  =/=  Z
) ) )
4443exlimiv 1784 . . . . . . . . 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 1784 . . . . . . 7  |-  ( E. s  x R s  ->  ( E. t
( x R t  <-> 
t  =/=  Z )  ->  E. y ( x R y  /\  y  =/=  Z ) ) )
472, 46sylbi 200 . . . . . 6  |-  ( E. t  x R t  ->  ( E. t
( x R t  <-> 
t  =/=  Z )  ->  E. y ( x R y  /\  y  =/=  Z ) ) )
4847imp 436 . . . . 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 3488 . . 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 6937 . . 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 6942 . . . 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 3461 . 2  |-  ( ( R  e.  V  /\  Z  e.  W )  ->  ( R supp  Z ) 
C_  ( `' R " ( _V  \  { Z } ) ) )
55 suppimacnvss 6943 . 2  |-  ( ( R  e.  V  /\  Z  e.  W )  ->  ( `' R "
( _V  \  { Z } ) )  C_  ( R supp  Z )
)
5654, 55eqssd 3435 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 189    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904   {cab 2457    =/= wne 2641   _Vcvv 3031    \ cdif 3387   {csn 3959   class class class wbr 4395   `'ccnv 4838   "cima 4842  (class class class)co 6308   supp csupp 6933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-supp 6934
This theorem is referenced by:  frnsuppeq  6945  suppun  6954  mptsuppdifd  6956  supp0cosupp0  6973  imacosupp  6974  fdmfisuppfi  7910  fsuppun  7920  fsuppco  7933  gsumval3a  17615  gsumzf1o  17624  gsumzaddlem  17632  gsumzmhm  17648  gsumzoppg  17655  deg1val  23124  suppss3  28387  ffsrn  28389  fpwrelmapffslem  28392  sitgclg  29248  eulerpartlemmf  29281  eulerpartlemgf  29285  fidmfisupp  37550
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