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Theorem fnsuppres 6949
Description: Two ways to express restriction of a support set. (Contributed by Stefan O'Rear, 5-Feb-2015.) (Revised by AV, 28-May-2019.)
Assertion
Ref Expression
fnsuppres  |-  ( ( F  Fn  ( A  u.  B )  /\  ( F  e.  W  /\  Z  e.  V
)  /\  ( A  i^i  B )  =  (/) )  ->  ( ( F supp 
Z )  C_  A  <->  ( F  |`  B )  =  ( B  X.  { Z } ) ) )

Proof of Theorem fnsuppres
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 fndm 5689 . . . . . 6  |-  ( F  Fn  ( A  u.  B )  ->  dom  F  =  ( A  u.  B ) )
2 rabeq 3074 . . . . . 6  |-  ( dom 
F  =  ( A  u.  B )  ->  { a  e.  dom  F  |  ( F `  a )  =/=  Z }  =  { a  e.  ( A  u.  B
)  |  ( F `
 a )  =/= 
Z } )
31, 2syl 17 . . . . 5  |-  ( F  Fn  ( A  u.  B )  ->  { a  e.  dom  F  | 
( F `  a
)  =/=  Z }  =  { a  e.  ( A  u.  B )  |  ( F `  a )  =/=  Z } )
433ad2ant1 1026 . . . 4  |-  ( ( F  Fn  ( A  u.  B )  /\  ( F  e.  W  /\  Z  e.  V
)  /\  ( A  i^i  B )  =  (/) )  ->  { a  e. 
dom  F  |  ( F `  a )  =/=  Z }  =  {
a  e.  ( A  u.  B )  |  ( F `  a
)  =/=  Z }
)
54sseq1d 3491 . . 3  |-  ( ( F  Fn  ( A  u.  B )  /\  ( F  e.  W  /\  Z  e.  V
)  /\  ( A  i^i  B )  =  (/) )  ->  ( { a  e.  dom  F  | 
( F `  a
)  =/=  Z }  C_  A  <->  { a  e.  ( A  u.  B )  |  ( F `  a )  =/=  Z }  C_  A ) )
6 unss 3640 . . . . 5  |-  ( ( { 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
)
7 ssrab2 3546 . . . . . 6  |-  { a  e.  A  |  ( F `  a )  =/=  Z }  C_  A
87biantrur 508 . . . . 5  |-  ( { a  e.  B  | 
( F `  a
)  =/=  Z }  C_  A  <->  ( { a  e.  A  |  ( F `  a )  =/=  Z }  C_  A  /\  { a  e.  B  |  ( F `
 a )  =/= 
Z }  C_  A
) )
9 rabun2 3752 . . . . . 6  |-  { a  e.  ( A  u.  B )  |  ( F `  a )  =/=  Z }  =  ( { a  e.  A  |  ( F `  a )  =/=  Z }  u.  { a  e.  B  |  ( F `  a )  =/=  Z } )
109sseq1i 3488 . . . . 5  |-  ( { 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 )
116, 8, 103bitr4ri 281 . . . 4  |-  ( { a  e.  ( A  u.  B )  |  ( F `  a
)  =/=  Z }  C_  A  <->  { a  e.  B  |  ( F `  a )  =/=  Z }  C_  A )
12 rabss 3538 . . . . 5  |-  ( { a  e.  B  | 
( F `  a
)  =/=  Z }  C_  A  <->  A. a  e.  B  ( ( F `  a )  =/=  Z  ->  a  e.  A ) )
13 fvres 5891 . . . . . . . . 9  |-  ( a  e.  B  ->  (
( F  |`  B ) `
 a )  =  ( F `  a
) )
1413adantl 467 . . . . . . . 8  |-  ( ( ( F  Fn  ( A  u.  B )  /\  ( F  e.  W  /\  Z  e.  V
)  /\  ( A  i^i  B )  =  (/) )  /\  a  e.  B
)  ->  ( ( F  |`  B ) `  a )  =  ( F `  a ) )
15 simp2r 1032 . . . . . . . . 9  |-  ( ( F  Fn  ( A  u.  B )  /\  ( F  e.  W  /\  Z  e.  V
)  /\  ( A  i^i  B )  =  (/) )  ->  Z  e.  V
)
16 fvconst2g 6129 . . . . . . . . 9  |-  ( ( Z  e.  V  /\  a  e.  B )  ->  ( ( B  X.  { Z } ) `  a )  =  Z )
1715, 16sylan 473 . . . . . . . 8  |-  ( ( ( F  Fn  ( A  u.  B )  /\  ( F  e.  W  /\  Z  e.  V
)  /\  ( A  i^i  B )  =  (/) )  /\  a  e.  B
)  ->  ( ( B  X.  { Z }
) `  a )  =  Z )
1814, 17eqeq12d 2444 . . . . . . 7  |-  ( ( ( F  Fn  ( A  u.  B )  /\  ( F  e.  W  /\  Z  e.  V
)  /\  ( A  i^i  B )  =  (/) )  /\  a  e.  B
)  ->  ( (
( F  |`  B ) `
 a )  =  ( ( B  X.  { Z } ) `  a )  <->  ( F `  a )  =  Z ) )
19 nne 2624 . . . . . . . 8  |-  ( -.  ( F `  a
)  =/=  Z  <->  ( F `  a )  =  Z )
2019a1i 11 . . . . . . 7  |-  ( ( ( F  Fn  ( A  u.  B )  /\  ( F  e.  W  /\  Z  e.  V
)  /\  ( A  i^i  B )  =  (/) )  /\  a  e.  B
)  ->  ( -.  ( F `  a )  =/=  Z  <->  ( F `  a )  =  Z ) )
21 id 23 . . . . . . . . 9  |-  ( a  e.  B  ->  a  e.  B )
22 simp3 1007 . . . . . . . . 9  |-  ( ( F  Fn  ( A  u.  B )  /\  ( F  e.  W  /\  Z  e.  V
)  /\  ( A  i^i  B )  =  (/) )  ->  ( A  i^i  B )  =  (/) )
23 minel 3848 . . . . . . . . 9  |-  ( ( a  e.  B  /\  ( A  i^i  B )  =  (/) )  ->  -.  a  e.  A )
2421, 22, 23syl2anr 480 . . . . . . . 8  |-  ( ( ( F  Fn  ( A  u.  B )  /\  ( F  e.  W  /\  Z  e.  V
)  /\  ( A  i^i  B )  =  (/) )  /\  a  e.  B
)  ->  -.  a  e.  A )
25 mtt 340 . . . . . . . 8  |-  ( -.  a  e.  A  -> 
( -.  ( F `
 a )  =/= 
Z  <->  ( ( F `
 a )  =/= 
Z  ->  a  e.  A ) ) )
2624, 25syl 17 . . . . . . 7  |-  ( ( ( F  Fn  ( A  u.  B )  /\  ( F  e.  W  /\  Z  e.  V
)  /\  ( A  i^i  B )  =  (/) )  /\  a  e.  B
)  ->  ( -.  ( F `  a )  =/=  Z  <->  ( ( F `  a )  =/=  Z  ->  a  e.  A ) ) )
2718, 20, 263bitr2rd 285 . . . . . 6  |-  ( ( ( F  Fn  ( A  u.  B )  /\  ( F  e.  W  /\  Z  e.  V
)  /\  ( A  i^i  B )  =  (/) )  /\  a  e.  B
)  ->  ( (
( F `  a
)  =/=  Z  -> 
a  e.  A )  <-> 
( ( F  |`  B ) `  a
)  =  ( ( B  X.  { Z } ) `  a
) ) )
2827ralbidva 2861 . . . . 5  |-  ( ( F  Fn  ( A  u.  B )  /\  ( F  e.  W  /\  Z  e.  V
)  /\  ( A  i^i  B )  =  (/) )  ->  ( A. a  e.  B  ( ( F `  a )  =/=  Z  ->  a  e.  A )  <->  A. a  e.  B  ( ( F  |`  B ) `  a )  =  ( ( B  X.  { Z } ) `  a
) ) )
2912, 28syl5bb 260 . . . 4  |-  ( ( F  Fn  ( A  u.  B )  /\  ( F  e.  W  /\  Z  e.  V
)  /\  ( A  i^i  B )  =  (/) )  ->  ( { a  e.  B  |  ( F `  a )  =/=  Z }  C_  A 
<-> 
A. a  e.  B  ( ( F  |`  B ) `  a
)  =  ( ( B  X.  { Z } ) `  a
) ) )
3011, 29syl5bb 260 . . 3  |-  ( ( F  Fn  ( A  u.  B )  /\  ( F  e.  W  /\  Z  e.  V
)  /\  ( A  i^i  B )  =  (/) )  ->  ( { a  e.  ( A  u.  B )  |  ( F `  a )  =/=  Z }  C_  A 
<-> 
A. a  e.  B  ( ( F  |`  B ) `  a
)  =  ( ( B  X.  { Z } ) `  a
) ) )
315, 30bitrd 256 . 2  |-  ( ( F  Fn  ( A  u.  B )  /\  ( F  e.  W  /\  Z  e.  V
)  /\  ( A  i^i  B )  =  (/) )  ->  ( { a  e.  dom  F  | 
( F `  a
)  =/=  Z }  C_  A  <->  A. a  e.  B  ( ( F  |`  B ) `  a
)  =  ( ( B  X.  { Z } ) `  a
) ) )
32 fnfun 5687 . . . . . . 7  |-  ( F  Fn  ( A  u.  B )  ->  Fun  F )
33323anim1i 1191 . . . . . 6  |-  ( ( F  Fn  ( A  u.  B )  /\  F  e.  W  /\  Z  e.  V )  ->  ( Fun  F  /\  F  e.  W  /\  Z  e.  V )
)
34333expb 1206 . . . . 5  |-  ( ( F  Fn  ( A  u.  B )  /\  ( F  e.  W  /\  Z  e.  V
) )  ->  ( Fun  F  /\  F  e.  W  /\  Z  e.  V ) )
35 suppval1 6927 . . . . 5  |-  ( ( Fun  F  /\  F  e.  W  /\  Z  e.  V )  ->  ( F supp  Z )  =  {
a  e.  dom  F  |  ( F `  a )  =/=  Z } )
3634, 35syl 17 . . . 4  |-  ( ( F  Fn  ( A  u.  B )  /\  ( F  e.  W  /\  Z  e.  V
) )  ->  ( F supp  Z )  =  {
a  e.  dom  F  |  ( F `  a )  =/=  Z } )
37363adant3 1025 . . 3  |-  ( ( F  Fn  ( A  u.  B )  /\  ( F  e.  W  /\  Z  e.  V
)  /\  ( A  i^i  B )  =  (/) )  ->  ( F supp  Z
)  =  { a  e.  dom  F  | 
( F `  a
)  =/=  Z }
)
3837sseq1d 3491 . 2  |-  ( ( F  Fn  ( A  u.  B )  /\  ( F  e.  W  /\  Z  e.  V
)  /\  ( A  i^i  B )  =  (/) )  ->  ( ( F supp 
Z )  C_  A  <->  { a  e.  dom  F  |  ( F `  a )  =/=  Z }  C_  A ) )
39 simp1 1005 . . . 4  |-  ( ( F  Fn  ( A  u.  B )  /\  ( F  e.  W  /\  Z  e.  V
)  /\  ( A  i^i  B )  =  (/) )  ->  F  Fn  ( A  u.  B )
)
40 ssun2 3630 . . . . 5  |-  B  C_  ( A  u.  B
)
4140a1i 11 . . . 4  |-  ( ( F  Fn  ( A  u.  B )  /\  ( F  e.  W  /\  Z  e.  V
)  /\  ( A  i^i  B )  =  (/) )  ->  B  C_  ( A  u.  B )
)
42 fnssres 5703 . . . 4  |-  ( ( F  Fn  ( A  u.  B )  /\  B  C_  ( A  u.  B ) )  -> 
( F  |`  B )  Fn  B )
4339, 41, 42syl2anc 665 . . 3  |-  ( ( F  Fn  ( A  u.  B )  /\  ( F  e.  W  /\  Z  e.  V
)  /\  ( A  i^i  B )  =  (/) )  ->  ( F  |`  B )  Fn  B
)
44 fnconstg 5784 . . . . 5  |-  ( Z  e.  V  ->  ( B  X.  { Z }
)  Fn  B )
4544adantl 467 . . . 4  |-  ( ( F  e.  W  /\  Z  e.  V )  ->  ( B  X.  { Z } )  Fn  B
)
46453ad2ant2 1027 . . 3  |-  ( ( F  Fn  ( A  u.  B )  /\  ( F  e.  W  /\  Z  e.  V
)  /\  ( A  i^i  B )  =  (/) )  ->  ( B  X.  { Z } )  Fn  B )
47 eqfnfv 5987 . . 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
) ) )
4843, 46, 47syl2anc 665 . 2  |-  ( ( F  Fn  ( A  u.  B )  /\  ( F  e.  W  /\  Z  e.  V
)  /\  ( A  i^i  B )  =  (/) )  ->  ( ( F  |`  B )  =  ( B  X.  { Z } )  <->  A. a  e.  B  ( ( F  |`  B ) `  a )  =  ( ( B  X.  { Z } ) `  a
) ) )
4931, 38, 483bitr4d 288 1  |-  ( ( F  Fn  ( A  u.  B )  /\  ( F  e.  W  /\  Z  e.  V
)  /\  ( A  i^i  B )  =  (/) )  ->  ( ( F supp 
Z )  C_  A  <->  ( F  |`  B )  =  ( B  X.  { Z } ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868    =/= wne 2618   A.wral 2775   {crab 2779    u. cun 3434    i^i cin 3435    C_ wss 3436   (/)c0 3761   {csn 3996    X. cxp 4847   dom cdm 4849    |` cres 4851   Fun wfun 5591    Fn wfn 5592   ` cfv 5597  (class class class)co 6301   supp csupp 6921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-fv 5605  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-supp 6922
This theorem is referenced by:  fnsuppeq0  6950  frlmsslss2  19319  resf1o  28308
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