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Theorem fnsuppresOLD 6132
Description: Two ways to express restriction of a support set. (Contributed by Stefan O'Rear, 5-Feb-2015.) Obsolete version of fnsuppres 6945 as of 28-May-2019. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
fnsuppresOLD  |-  ( ( 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 fnsuppresOLD
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 unss 3674 . . . 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 3581 . . . . 5  |-  { a  e.  A  |  ( F `  a )  =/=  Z }  C_  A
32biantrur 506 . . . 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 3784 . . . . 5  |-  { a  e.  ( A  u.  B )  |  ( F `  a )  =/=  Z }  =  ( { a  e.  A  |  ( F `  a )  =/=  Z }  u.  { a  e.  B  |  ( F `  a )  =/=  Z } )
54sseq1i 3523 . . . 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 278 . . 3  |-  ( { a  e.  ( A  u.  B )  |  ( F `  a
)  =/=  Z }  C_  A  <->  { a  e.  B  |  ( F `  a )  =/=  Z }  C_  A )
7 rabss 3573 . . . 4  |-  ( { a  e.  B  | 
( F `  a
)  =/=  Z }  C_  A  <->  A. a  e.  B  ( ( F `  a )  =/=  Z  ->  a  e.  A ) )
8 fvres 5886 . . . . . . . 8  |-  ( a  e.  B  ->  (
( F  |`  B ) `
 a )  =  ( F `  a
) )
98adantl 466 . . . . . . 7  |-  ( ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B
)  =  (/)  /\  Z  e.  V )  /\  a  e.  B )  ->  (
( F  |`  B ) `
 a )  =  ( F `  a
) )
10 fvconst2g 6126 . . . . . . . 8  |-  ( ( Z  e.  V  /\  a  e.  B )  ->  ( ( B  X.  { Z } ) `  a )  =  Z )
11103ad2antl3 1160 . . . . . . 7  |-  ( ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B
)  =  (/)  /\  Z  e.  V )  /\  a  e.  B )  ->  (
( B  X.  { Z } ) `  a
)  =  Z )
129, 11eqeq12d 2479 . . . . . 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 2658 . . . . . . 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 22 . . . . . . . 8  |-  ( a  e.  B  ->  a  e.  B )
16 simp2 997 . . . . . . . 8  |-  ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B )  =  (/)  /\  Z  e.  V )  ->  ( A  i^i  B )  =  (/) )
17 minel 3885 . . . . . . . 8  |-  ( ( a  e.  B  /\  ( A  i^i  B )  =  (/) )  ->  -.  a  e.  A )
1815, 16, 17syl2anr 478 . . . . . . 7  |-  ( ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B
)  =  (/)  /\  Z  e.  V )  /\  a  e.  B )  ->  -.  a  e.  A )
19 mtt 339 . . . . . . 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 282 . . . . 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 2893 . . . 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 257 . . 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 257 . 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 fnniniseg2OLD 6012 . . . 4  |-  ( F  Fn  ( A  u.  B )  ->  ( `' F " ( _V 
\  { Z }
) )  =  {
a  e.  ( A  u.  B )  |  ( F `  a
)  =/=  Z }
)
26253ad2ant1 1017 . . 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 3526 . 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 996 . . . 4  |-  ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B )  =  (/)  /\  Z  e.  V )  ->  F  Fn  ( A  u.  B
) )
29 ssun2 3664 . . . . 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 5700 . . . 4  |-  ( ( F  Fn  ( A  u.  B )  /\  B  C_  ( A  u.  B ) )  -> 
( F  |`  B )  Fn  B )
3228, 30, 31syl2anc 661 . . 3  |-  ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B )  =  (/)  /\  Z  e.  V )  ->  ( F  |`  B )  Fn  B )
33 fnconstg 5779 . . . 4  |-  ( Z  e.  V  ->  ( B  X.  { Z }
)  Fn  B )
34333ad2ant3 1019 . . 3  |-  ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B )  =  (/)  /\  Z  e.  V )  ->  ( B  X.  { Z }
)  Fn  B )
35 eqfnfv 5982 . . 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 661 . 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 285 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   {crab 2811   _Vcvv 3109    \ cdif 3468    u. cun 3469    i^i cin 3470    C_ wss 3471   (/)c0 3793   {csn 4032    X. cxp 5006   `'ccnv 5007    |` cres 5010   "cima 5011    Fn wfn 5589   ` cfv 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602
This theorem is referenced by:  fnsuppeq0OLD  6133  frlmsslss2OLD  18933
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