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Theorem ressuppfi 7733
Description: If the support of the restriction of a function by a set which, subtracted from the domain of the function so that its difference is finite, the support of the function itself is finite. (Contributed by AV, 22-Apr-2019.)
Hypotheses
Ref Expression
ressuppfi.b  |-  ( ph  ->  ( dom  F  \  B )  e.  Fin )
ressuppfi.f  |-  ( ph  ->  F  e.  W )
ressuppfi.g  |-  ( ph  ->  G  =  ( F  |`  B ) )
ressuppfi.s  |-  ( ph  ->  ( G supp  Z )  e.  Fin )
ressuppfi.z  |-  ( ph  ->  Z  e.  V )
Assertion
Ref Expression
ressuppfi  |-  ( ph  ->  ( F supp  Z )  e.  Fin )

Proof of Theorem ressuppfi
StepHypRef Expression
1 ressuppfi.g . . . . . 6  |-  ( ph  ->  G  =  ( F  |`  B ) )
21eqcomd 2457 . . . . 5  |-  ( ph  ->  ( F  |`  B )  =  G )
32oveq1d 6191 . . . 4  |-  ( ph  ->  ( ( F  |`  B ) supp  Z )  =  ( G supp  Z
) )
4 ressuppfi.s . . . 4  |-  ( ph  ->  ( G supp  Z )  e.  Fin )
53, 4eqeltrd 2536 . . 3  |-  ( ph  ->  ( ( F  |`  B ) supp  Z )  e.  Fin )
6 ressuppfi.b . . 3  |-  ( ph  ->  ( dom  F  \  B )  e.  Fin )
7 unfi 7666 . . 3  |-  ( ( ( ( F  |`  B ) supp  Z )  e.  Fin  /\  ( dom 
F  \  B )  e.  Fin )  ->  (
( ( F  |`  B ) supp  Z )  u.  ( dom  F  \  B ) )  e. 
Fin )
85, 6, 7syl2anc 661 . 2  |-  ( ph  ->  ( ( ( F  |`  B ) supp  Z )  u.  ( dom  F  \  B ) )  e. 
Fin )
9 ressuppfi.f . . 3  |-  ( ph  ->  F  e.  W )
10 ressuppfi.z . . 3  |-  ( ph  ->  Z  e.  V )
11 ressuppssdif 6796 . . 3  |-  ( ( F  e.  W  /\  Z  e.  V )  ->  ( F supp  Z ) 
C_  ( ( ( F  |`  B ) supp  Z )  u.  ( dom 
F  \  B )
) )
129, 10, 11syl2anc 661 . 2  |-  ( ph  ->  ( F supp  Z ) 
C_  ( ( ( F  |`  B ) supp  Z )  u.  ( dom 
F  \  B )
) )
13 ssfi 7620 . 2  |-  ( ( ( ( ( F  |`  B ) supp  Z )  u.  ( dom  F  \  B ) )  e. 
Fin  /\  ( F supp  Z )  C_  ( (
( F  |`  B ) supp 
Z )  u.  ( dom  F  \  B ) ) )  ->  ( F supp  Z )  e.  Fin )
148, 12, 13syl2anc 661 1  |-  ( ph  ->  ( F supp  Z )  e.  Fin )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1757    \ cdif 3409    u. cun 3410    C_ wss 3412   dom cdm 4924    |` cres 4926  (class class class)co 6176   supp csupp 6776   Fincfn 7396
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 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-reu 2799  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-int 4213  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-om 6563  df-supp 6777  df-recs 6918  df-rdg 6952  df-oadd 7010  df-er 7187  df-en 7397  df-fin 7400
This theorem is referenced by:  resfsupp  7734
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