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Theorem fnfi 7676
Description: A version of fnex 6029 for finite sets that does not require Replacement. (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.)
Assertion
Ref Expression
fnfi  |-  ( ( F  Fn  A  /\  A  e.  Fin )  ->  F  e.  Fin )

Proof of Theorem fnfi
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fnresdm 5604 . . 3  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
21adantr 465 . 2  |-  ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  A )  =  F )
3 reseq2 5189 . . . . . 6  |-  ( x  =  (/)  ->  ( F  |`  x )  =  ( F  |`  (/) ) )
43eleq1d 2518 . . . . 5  |-  ( x  =  (/)  ->  ( ( F  |`  x )  e.  Fin  <->  ( F  |`  (/) )  e.  Fin )
)
54imbi2d 316 . . . 4  |-  ( x  =  (/)  ->  ( ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  x )  e.  Fin ) 
<->  ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  (/) )  e.  Fin )
) )
6 reseq2 5189 . . . . . 6  |-  ( x  =  y  ->  ( F  |`  x )  =  ( F  |`  y
) )
76eleq1d 2518 . . . . 5  |-  ( x  =  y  ->  (
( F  |`  x
)  e.  Fin  <->  ( F  |`  y )  e.  Fin ) )
87imbi2d 316 . . . 4  |-  ( x  =  y  ->  (
( ( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  x )  e.  Fin ) 
<->  ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  y )  e.  Fin ) ) )
9 reseq2 5189 . . . . . 6  |-  ( x  =  ( y  u. 
{ z } )  ->  ( F  |`  x )  =  ( F  |`  ( y  u.  { z } ) ) )
109eleq1d 2518 . . . . 5  |-  ( x  =  ( y  u. 
{ z } )  ->  ( ( F  |`  x )  e.  Fin  <->  ( F  |`  ( y  u. 
{ z } ) )  e.  Fin )
)
1110imbi2d 316 . . . 4  |-  ( x  =  ( y  u. 
{ z } )  ->  ( ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  x
)  e.  Fin )  <->  ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  ( y  u.  {
z } ) )  e.  Fin ) ) )
12 reseq2 5189 . . . . . 6  |-  ( x  =  A  ->  ( F  |`  x )  =  ( F  |`  A ) )
1312eleq1d 2518 . . . . 5  |-  ( x  =  A  ->  (
( F  |`  x
)  e.  Fin  <->  ( F  |`  A )  e.  Fin ) )
1413imbi2d 316 . . . 4  |-  ( x  =  A  ->  (
( ( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  x )  e.  Fin ) 
<->  ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  A )  e.  Fin ) ) )
15 res0 5199 . . . . . 6  |-  ( F  |`  (/) )  =  (/)
16 0fin 7627 . . . . . 6  |-  (/)  e.  Fin
1715, 16eqeltri 2532 . . . . 5  |-  ( F  |`  (/) )  e.  Fin
1817a1i 11 . . . 4  |-  ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  (/) )  e. 
Fin )
19 resundi 5208 . . . . . . . 8  |-  ( F  |`  ( y  u.  {
z } ) )  =  ( ( F  |`  y )  u.  ( F  |`  { z } ) )
20 snfi 7476 . . . . . . . . . 10  |-  { <. z ,  ( F `  z ) >. }  e.  Fin
21 fnfun 5592 . . . . . . . . . . . 12  |-  ( F  Fn  A  ->  Fun  F )
22 funressn 5980 . . . . . . . . . . . 12  |-  ( Fun 
F  ->  ( F  |` 
{ z } ) 
C_  { <. z ,  ( F `  z ) >. } )
2321, 22syl 16 . . . . . . . . . . 11  |-  ( F  Fn  A  ->  ( F  |`  { z } )  C_  { <. z ,  ( F `  z ) >. } )
2423adantr 465 . . . . . . . . . 10  |-  ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  { z } )  C_  { <. z ,  ( F `  z ) >. } )
25 ssfi 7620 . . . . . . . . . 10  |-  ( ( { <. z ,  ( F `  z )
>. }  e.  Fin  /\  ( F  |`  { z } )  C_  { <. z ,  ( F `  z ) >. } )  ->  ( F  |`  { z } )  e.  Fin )
2620, 24, 25sylancr 663 . . . . . . . . 9  |-  ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  { z } )  e.  Fin )
27 unfi 7666 . . . . . . . . 9  |-  ( ( ( F  |`  y
)  e.  Fin  /\  ( F  |`  { z } )  e.  Fin )  ->  ( ( F  |`  y )  u.  ( F  |`  { z } ) )  e.  Fin )
2826, 27sylan2 474 . . . . . . . 8  |-  ( ( ( F  |`  y
)  e.  Fin  /\  ( F  Fn  A  /\  A  e.  Fin ) )  ->  (
( F  |`  y
)  u.  ( F  |`  { z } ) )  e.  Fin )
2919, 28syl5eqel 2540 . . . . . . 7  |-  ( ( ( F  |`  y
)  e.  Fin  /\  ( F  Fn  A  /\  A  e.  Fin ) )  ->  ( F  |`  ( y  u. 
{ z } ) )  e.  Fin )
3029expcom 435 . . . . . 6  |-  ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( ( F  |`  y )  e.  Fin  ->  ( F  |`  (
y  u.  { z } ) )  e. 
Fin ) )
3130a2i 13 . . . . 5  |-  ( ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  y )  e.  Fin )  ->  ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  ( y  u. 
{ z } ) )  e.  Fin )
)
3231a1i 11 . . . 4  |-  ( y  e.  Fin  ->  (
( ( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  y )  e.  Fin )  ->  ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  ( y  u. 
{ z } ) )  e.  Fin )
) )
335, 8, 11, 14, 18, 32findcard2 7639 . . 3  |-  ( A  e.  Fin  ->  (
( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  A )  e.  Fin ) )
3433anabsi7 815 . 2  |-  ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  A )  e.  Fin )
352, 34eqeltrrd 2537 1  |-  ( ( F  Fn  A  /\  A  e.  Fin )  ->  F  e.  Fin )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1757    u. cun 3410    C_ wss 3412   (/)c0 3721   {csn 3961   <.cop 3967    |` cres 4926   Fun wfun 5496    Fn wfn 5497   ` cfv 5502   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-recs 6918  df-rdg 6952  df-1o 7006  df-oadd 7010  df-er 7187  df-en 7397  df-fin 7400
This theorem is referenced by:  unirnffid  7690  mptfi  7697  seqf1olem2  11933  seqf1o  11934  hashfzdm  12290  hashfirdm  12292  iswrd  12325  wrdfin  12336  isstruct2  14271  xpsfrnel  14589  psgnran  16109  usgrafilem2  23446  ftc1anclem3  28593  sstotbnd2  28797  prdstotbnd  28817  stoweidlem59  29978  resfnfinfin  30289
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