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Theorem fnsuppres 5981
Description: Two ways to express restriction of a support set. (Contributed by Stefan O'Rear, 5-Feb-2015.)
Assertion
Ref Expression
fnsuppres  |-  ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B )  =  (/)  /\  Z  e.  V )  ->  (
( `' F "
( _V  \  { Z } ) )  C_  A 
<->  ( F  |`  B )  =  ( B  X.  { Z } ) ) )

Proof of Theorem fnsuppres
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 unss 3507 . . . 4  |-  ( ( { a  e.  A  |  ( F `  a )  =/=  Z }  C_  A  /\  {
a  e.  B  | 
( F `  a
)  =/=  Z }  C_  A )  <->  ( {
a  e.  A  | 
( F `  a
)  =/=  Z }  u.  { a  e.  B  |  ( F `  a )  =/=  Z } )  C_  A
)
2 ssrab2 3414 . . . . 5  |-  { a  e.  A  |  ( F `  a )  =/=  Z }  C_  A
32biantrur 494 . . . 4  |-  ( { a  e.  B  | 
( F `  a
)  =/=  Z }  C_  A  <->  ( { a  e.  A  |  ( F `  a )  =/=  Z }  C_  A  /\  { a  e.  B  |  ( F `
 a )  =/= 
Z }  C_  A
) )
4 rabun2 3605 . . . . 5  |-  { a  e.  ( A  u.  B )  |  ( F `  a )  =/=  Z }  =  ( { a  e.  A  |  ( F `  a )  =/=  Z }  u.  { a  e.  B  |  ( F `  a )  =/=  Z } )
54sseq1i 3358 . . . 4  |-  ( { a  e.  ( A  u.  B )  |  ( F `  a
)  =/=  Z }  C_  A  <->  ( { a  e.  A  |  ( F `  a )  =/=  Z }  u.  { a  e.  B  | 
( F `  a
)  =/=  Z }
)  C_  A )
61, 3, 53bitr4ri 271 . . 3  |-  ( { a  e.  ( A  u.  B )  |  ( F `  a
)  =/=  Z }  C_  A  <->  { a  e.  B  |  ( F `  a )  =/=  Z }  C_  A )
7 rabss 3406 . . . 4  |-  ( { a  e.  B  | 
( F `  a
)  =/=  Z }  C_  A  <->  A. a  e.  B  ( ( F `  a )  =/=  Z  ->  a  e.  A ) )
8 fvres 5774 . . . . . . . 8  |-  ( a  e.  B  ->  (
( F  |`  B ) `
 a )  =  ( F `  a
) )
98adantl 454 . . . . . . 7  |-  ( ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B
)  =  (/)  /\  Z  e.  V )  /\  a  e.  B )  ->  (
( F  |`  B ) `
 a )  =  ( F `  a
) )
10 fvconst2g 5974 . . . . . . . 8  |-  ( ( Z  e.  V  /\  a  e.  B )  ->  ( ( B  X.  { Z } ) `  a )  =  Z )
11103ad2antl3 1122 . . . . . . 7  |-  ( ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B
)  =  (/)  /\  Z  e.  V )  /\  a  e.  B )  ->  (
( B  X.  { Z } ) `  a
)  =  Z )
129, 11eqeq12d 2456 . . . . . 6  |-  ( ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B
)  =  (/)  /\  Z  e.  V )  /\  a  e.  B )  ->  (
( ( F  |`  B ) `  a
)  =  ( ( B  X.  { Z } ) `  a
)  <->  ( F `  a )  =  Z ) )
13 nne 2611 . . . . . . 7  |-  ( -.  ( F `  a
)  =/=  Z  <->  ( F `  a )  =  Z )
1413a1i 11 . . . . . 6  |-  ( ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B
)  =  (/)  /\  Z  e.  V )  /\  a  e.  B )  ->  ( -.  ( F `  a
)  =/=  Z  <->  ( F `  a )  =  Z ) )
15 id 21 . . . . . . . 8  |-  ( a  e.  B  ->  a  e.  B )
16 simp2 959 . . . . . . . 8  |-  ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B )  =  (/)  /\  Z  e.  V )  ->  ( A  i^i  B )  =  (/) )
17 minel 3707 . . . . . . . 8  |-  ( ( a  e.  B  /\  ( A  i^i  B )  =  (/) )  ->  -.  a  e.  A )
1815, 16, 17syl2anr 466 . . . . . . 7  |-  ( ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B
)  =  (/)  /\  Z  e.  V )  /\  a  e.  B )  ->  -.  a  e.  A )
19 mtt 331 . . . . . . 7  |-  ( -.  a  e.  A  -> 
( -.  ( F `
 a )  =/= 
Z  <->  ( ( F `
 a )  =/= 
Z  ->  a  e.  A ) ) )
2018, 19syl 16 . . . . . 6  |-  ( ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B
)  =  (/)  /\  Z  e.  V )  /\  a  e.  B )  ->  ( -.  ( F `  a
)  =/=  Z  <->  ( ( F `  a )  =/=  Z  ->  a  e.  A ) ) )
2112, 14, 203bitr2rd 275 . . . . 5  |-  ( ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B
)  =  (/)  /\  Z  e.  V )  /\  a  e.  B )  ->  (
( ( F `  a )  =/=  Z  ->  a  e.  A )  <-> 
( ( F  |`  B ) `  a
)  =  ( ( B  X.  { Z } ) `  a
) ) )
2221ralbidva 2727 . . . 4  |-  ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B )  =  (/)  /\  Z  e.  V )  ->  ( A. a  e.  B  ( ( F `  a )  =/=  Z  ->  a  e.  A )  <->  A. a  e.  B  ( ( F  |`  B ) `  a
)  =  ( ( B  X.  { Z } ) `  a
) ) )
237, 22syl5bb 250 . . 3  |-  ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B )  =  (/)  /\  Z  e.  V )  ->  ( { a  e.  B  |  ( F `  a )  =/=  Z }  C_  A  <->  A. a  e.  B  ( ( F  |`  B ) `  a )  =  ( ( B  X.  { Z } ) `  a
) ) )
246, 23syl5bb 250 . 2  |-  ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B )  =  (/)  /\  Z  e.  V )  ->  ( { a  e.  ( A  u.  B )  |  ( F `  a )  =/=  Z }  C_  A  <->  A. a  e.  B  ( ( F  |`  B ) `  a )  =  ( ( B  X.  { Z } ) `  a
) ) )
25 fnniniseg2 5883 . . . 4  |-  ( F  Fn  ( A  u.  B )  ->  ( `' F " ( _V 
\  { Z }
) )  =  {
a  e.  ( A  u.  B )  |  ( F `  a
)  =/=  Z }
)
26253ad2ant1 979 . . 3  |-  ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B )  =  (/)  /\  Z  e.  V )  ->  ( `' F " ( _V 
\  { Z }
) )  =  {
a  e.  ( A  u.  B )  |  ( F `  a
)  =/=  Z }
)
2726sseq1d 3361 . 2  |-  ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B )  =  (/)  /\  Z  e.  V )  ->  (
( `' F "
( _V  \  { Z } ) )  C_  A 
<->  { a  e.  ( A  u.  B )  |  ( F `  a )  =/=  Z }  C_  A ) )
28 simp1 958 . . . 4  |-  ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B )  =  (/)  /\  Z  e.  V )  ->  F  Fn  ( A  u.  B
) )
29 ssun2 3497 . . . . 5  |-  B  C_  ( A  u.  B
)
3029a1i 11 . . . 4  |-  ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B )  =  (/)  /\  Z  e.  V )  ->  B  C_  ( A  u.  B
) )
31 fnssres 5587 . . . 4  |-  ( ( F  Fn  ( A  u.  B )  /\  B  C_  ( A  u.  B ) )  -> 
( F  |`  B )  Fn  B )
3228, 30, 31syl2anc 644 . . 3  |-  ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B )  =  (/)  /\  Z  e.  V )  ->  ( F  |`  B )  Fn  B )
33 fnconstg 5660 . . . 4  |-  ( Z  e.  V  ->  ( B  X.  { Z }
)  Fn  B )
34333ad2ant3 981 . . 3  |-  ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B )  =  (/)  /\  Z  e.  V )  ->  ( B  X.  { Z }
)  Fn  B )
35 eqfnfv 5856 . . 3  |-  ( ( ( F  |`  B )  Fn  B  /\  ( B  X.  { Z }
)  Fn  B )  ->  ( ( F  |`  B )  =  ( B  X.  { Z } )  <->  A. a  e.  B  ( ( F  |`  B ) `  a )  =  ( ( B  X.  { Z } ) `  a
) ) )
3632, 34, 35syl2anc 644 . 2  |-  ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B )  =  (/)  /\  Z  e.  V )  ->  (
( F  |`  B )  =  ( B  X.  { Z } )  <->  A. a  e.  B  ( ( F  |`  B ) `  a )  =  ( ( B  X.  { Z } ) `  a
) ) )
3724, 27, 363bitr4d 278 1  |-  ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B )  =  (/)  /\  Z  e.  V )  ->  (
( `' F "
( _V  \  { Z } ) )  C_  A 
<->  ( F  |`  B )  =  ( B  X.  { Z } ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1727    =/= wne 2605   A.wral 2711   {crab 2715   _Vcvv 2962    \ cdif 3303    u. cun 3304    i^i cin 3305    C_ wss 3306   (/)c0 3613   {csn 3838    X. cxp 4905   `'ccnv 4906    |` cres 4909   "cima 4910    Fn wfn 5478   ` cfv 5483
This theorem is referenced by:  fnsuppeq0  5982  frlmsslss2  27260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-fv 5491
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