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Theorem ressuppfi 7847
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 2462 . . . . 5  |-  ( ph  ->  ( F  |`  B )  =  G )
32oveq1d 6285 . . . 4  |-  ( ph  ->  ( ( F  |`  B ) supp  Z )  =  ( G supp  Z
) )
4 ressuppfi.s . . . 4  |-  ( ph  ->  ( G supp  Z )  e.  Fin )
53, 4eqeltrd 2542 . . 3  |-  ( ph  ->  ( ( F  |`  B ) supp  Z )  e.  Fin )
6 ressuppfi.b . . 3  |-  ( ph  ->  ( dom  F  \  B )  e.  Fin )
7 unfi 7779 . . 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 659 . 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 6913 . . 3  |-  ( ( F  e.  W  /\  Z  e.  V )  ->  ( F supp  Z ) 
C_  ( ( ( F  |`  B ) supp  Z )  u.  ( dom 
F  \  B )
) )
129, 10, 11syl2anc 659 . 2  |-  ( ph  ->  ( F supp  Z ) 
C_  ( ( ( F  |`  B ) supp  Z )  u.  ( dom 
F  \  B )
) )
13 ssfi 7733 . 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 659 1  |-  ( ph  ->  ( F supp  Z )  e.  Fin )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823    \ cdif 3458    u. cun 3459    C_ wss 3461   dom cdm 4988    |` cres 4990  (class class class)co 6270   supp csupp 6891   Fincfn 7509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-supp 6892  df-recs 7034  df-rdg 7068  df-oadd 7126  df-er 7303  df-en 7510  df-fin 7513
This theorem is referenced by:  resfsupp  7848
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