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Theorem fnsuppres 5815
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 3425 . . . 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 3334 . . . . 5  |-  { a  e.  A  |  ( F `  a )  =/=  Z }  C_  A
32biantrur 492 . . . 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 3523 . . . . 5  |-  { a  e.  ( A  u.  B )  |  ( F `  a )  =/=  Z }  =  ( { a  e.  A  |  ( F `  a )  =/=  Z }  u.  { a  e.  B  |  ( F `  a )  =/=  Z } )
54sseq1i 3278 . . . 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 269 . . 3  |-  ( { a  e.  ( A  u.  B )  |  ( F `  a
)  =/=  Z }  C_  A  <->  { a  e.  B  |  ( F `  a )  =/=  Z }  C_  A )
7 rabss 3326 . . . 4  |-  ( { a  e.  B  | 
( F `  a
)  =/=  Z }  C_  A  <->  A. a  e.  B  ( ( F `  a )  =/=  Z  ->  a  e.  A ) )
8 fvres 5622 . . . . . . . 8  |-  ( a  e.  B  ->  (
( F  |`  B ) `
 a )  =  ( F `  a
) )
98adantl 452 . . . . . . 7  |-  ( ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B
)  =  (/)  /\  Z  e.  V )  /\  a  e.  B )  ->  (
( F  |`  B ) `
 a )  =  ( F `  a
) )
10 fvconst2g 5808 . . . . . . . 8  |-  ( ( Z  e.  V  /\  a  e.  B )  ->  ( ( B  X.  { Z } ) `  a )  =  Z )
11103ad2antl3 1119 . . . . . . 7  |-  ( ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B
)  =  (/)  /\  Z  e.  V )  /\  a  e.  B )  ->  (
( B  X.  { Z } ) `  a
)  =  Z )
129, 11eqeq12d 2372 . . . . . 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 2525 . . . . . . 7  |-  ( -.  ( F `  a
)  =/=  Z  <->  ( F `  a )  =  Z )
1413a1i 10 . . . . . 6  |-  ( ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B
)  =  (/)  /\  Z  e.  V )  /\  a  e.  B )  ->  ( -.  ( F `  a
)  =/=  Z  <->  ( F `  a )  =  Z ) )
15 id 19 . . . . . . . 8  |-  ( a  e.  B  ->  a  e.  B )
16 simp2 956 . . . . . . . 8  |-  ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B )  =  (/)  /\  Z  e.  V )  ->  ( A  i^i  B )  =  (/) )
17 minel 3586 . . . . . . . 8  |-  ( ( a  e.  B  /\  ( A  i^i  B )  =  (/) )  ->  -.  a  e.  A )
1815, 16, 17syl2anr 464 . . . . . . 7  |-  ( ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B
)  =  (/)  /\  Z  e.  V )  /\  a  e.  B )  ->  -.  a  e.  A )
19 mtt 329 . . . . . . 7  |-  ( -.  a  e.  A  -> 
( -.  ( F `
 a )  =/= 
Z  <->  ( ( F `
 a )  =/= 
Z  ->  a  e.  A ) ) )
2018, 19syl 15 . . . . . 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 273 . . . . 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 2635 . . . 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 248 . . 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 248 . 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 5729 . . . 4  |-  ( F  Fn  ( A  u.  B )  ->  ( `' F " ( _V 
\  { Z }
) )  =  {
a  e.  ( A  u.  B )  |  ( F `  a
)  =/=  Z }
)
26253ad2ant1 976 . . 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 3281 . 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 955 . . . 4  |-  ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B )  =  (/)  /\  Z  e.  V )  ->  F  Fn  ( A  u.  B
) )
29 ssun2 3415 . . . . 5  |-  B  C_  ( A  u.  B
)
3029a1i 10 . . . 4  |-  ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B )  =  (/)  /\  Z  e.  V )  ->  B  C_  ( A  u.  B
) )
31 fnssres 5436 . . . 4  |-  ( ( F  Fn  ( A  u.  B )  /\  B  C_  ( A  u.  B ) )  -> 
( F  |`  B )  Fn  B )
3228, 30, 31syl2anc 642 . . 3  |-  ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B )  =  (/)  /\  Z  e.  V )  ->  ( F  |`  B )  Fn  B )
33 fnconstg 5509 . . . 4  |-  ( Z  e.  V  ->  ( B  X.  { Z }
)  Fn  B )
34333ad2ant3 978 . . 3  |-  ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B )  =  (/)  /\  Z  e.  V )  ->  ( B  X.  { Z }
)  Fn  B )
35 eqfnfv 5702 . . 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 642 . 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 276 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 176    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710    =/= wne 2521   A.wral 2619   {crab 2623   _Vcvv 2864    \ cdif 3225    u. cun 3226    i^i cin 3227    C_ wss 3228   (/)c0 3531   {csn 3716    X. cxp 4766   `'ccnv 4767    |` cres 4770   "cima 4771    Fn wfn 5329   ` cfv 5334
This theorem is referenced by:  fnsuppeq0  5816  frlmsslss2  26568
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-fv 5342
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