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Theorem fnsuppresOLD 6050
Description: Two ways to express restriction of a support set. (Contributed by Stefan O'Rear, 5-Feb-2015.) Obsolete version of fnsuppres 6829 as of 28-May-2019. (New usage 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 3641 . . . 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 3548 . . . . 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 3740 . . . . 5  |-  { a  e.  ( A  u.  B )  |  ( F `  a )  =/=  Z }  =  ( { a  e.  A  |  ( F `  a )  =/=  Z }  u.  { a  e.  B  |  ( F `  a )  =/=  Z } )
54sseq1i 3491 . . . 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 3540 . . . 4  |-  ( { a  e.  B  | 
( F `  a
)  =/=  Z }  C_  A  <->  A. a  e.  B  ( ( F `  a )  =/=  Z  ->  a  e.  A ) )
8 fvres 5816 . . . . . . . 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 6043 . . . . . . . 8  |-  ( ( Z  e.  V  /\  a  e.  B )  ->  ( ( B  X.  { Z } ) `  a )  =  Z )
11103ad2antl3 1152 . . . . . . 7  |-  ( ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B
)  =  (/)  /\  Z  e.  V )  /\  a  e.  B )  ->  (
( B  X.  { Z } ) `  a
)  =  Z )
129, 11eqeq12d 2476 . . . . . 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 2654 . . . . . . 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 989 . . . . . . . 8  |-  ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B )  =  (/)  /\  Z  e.  V )  ->  ( A  i^i  B )  =  (/) )
17 minel 3845 . . . . . . . 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 2844 . . . 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 5939 . . . 4  |-  ( F  Fn  ( A  u.  B )  ->  ( `' F " ( _V 
\  { Z }
) )  =  {
a  e.  ( A  u.  B )  |  ( F `  a
)  =/=  Z }
)
26253ad2ant1 1009 . . 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 3494 . 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 988 . . . 4  |-  ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B )  =  (/)  /\  Z  e.  V )  ->  F  Fn  ( A  u.  B
) )
29 ssun2 3631 . . . . 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 5635 . . . 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 5709 . . . 4  |-  ( Z  e.  V  ->  ( B  X.  { Z }
)  Fn  B )
34333ad2ant3 1011 . . 3  |-  ( ( F  Fn  ( A  u.  B )  /\  ( A  i^i  B )  =  (/)  /\  Z  e.  V )  ->  ( B  X.  { Z }
)  Fn  B )
35 eqfnfv 5909 . . 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 965    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   {crab 2803   _Vcvv 3078    \ cdif 3436    u. cun 3437    i^i cin 3438    C_ wss 3439   (/)c0 3748   {csn 3988    X. cxp 4949   `'ccnv 4950    |` cres 4953   "cima 4954    Fn wfn 5524   ` cfv 5529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fv 5537
This theorem is referenced by:  fnsuppeq0OLD  6051  frlmsslss2OLD  18328
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