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Theorem fnfi 7376
Description: A version of fnex 5953 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 5546 . . 3  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
21adantr 452 . 2  |-  ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  A )  =  F )
3 reseq2 5133 . . . . . 6  |-  ( x  =  (/)  ->  ( F  |`  x )  =  ( F  |`  (/) ) )
43eleq1d 2501 . . . . 5  |-  ( x  =  (/)  ->  ( ( F  |`  x )  e.  Fin  <->  ( F  |`  (/) )  e.  Fin )
)
54imbi2d 308 . . . 4  |-  ( x  =  (/)  ->  ( ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  x )  e.  Fin ) 
<->  ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  (/) )  e.  Fin )
) )
6 reseq2 5133 . . . . . 6  |-  ( x  =  y  ->  ( F  |`  x )  =  ( F  |`  y
) )
76eleq1d 2501 . . . . 5  |-  ( x  =  y  ->  (
( F  |`  x
)  e.  Fin  <->  ( F  |`  y )  e.  Fin ) )
87imbi2d 308 . . . 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 5133 . . . . . 6  |-  ( x  =  ( y  u. 
{ z } )  ->  ( F  |`  x )  =  ( F  |`  ( y  u.  { z } ) ) )
109eleq1d 2501 . . . . 5  |-  ( x  =  ( y  u. 
{ z } )  ->  ( ( F  |`  x )  e.  Fin  <->  ( F  |`  ( y  u. 
{ z } ) )  e.  Fin )
)
1110imbi2d 308 . . . 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 5133 . . . . . 6  |-  ( x  =  A  ->  ( F  |`  x )  =  ( F  |`  A ) )
1312eleq1d 2501 . . . . 5  |-  ( x  =  A  ->  (
( F  |`  x
)  e.  Fin  <->  ( F  |`  A )  e.  Fin ) )
1413imbi2d 308 . . . 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 5142 . . . . . 6  |-  ( F  |`  (/) )  =  (/)
16 0fin 7328 . . . . . 6  |-  (/)  e.  Fin
1715, 16eqeltri 2505 . . . . 5  |-  ( F  |`  (/) )  e.  Fin
1817a1i 11 . . . 4  |-  ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  (/) )  e. 
Fin )
19 resundi 5152 . . . . . . . 8  |-  ( F  |`  ( y  u.  {
z } ) )  =  ( ( F  |`  y )  u.  ( F  |`  { z } ) )
20 snfi 7179 . . . . . . . . . 10  |-  { <. z ,  ( F `  z ) >. }  e.  Fin
21 fnfun 5534 . . . . . . . . . . . 12  |-  ( F  Fn  A  ->  Fun  F )
22 funressn 5911 . . . . . . . . . . . 12  |-  ( Fun 
F  ->  ( F  |` 
{ z } ) 
C_  { <. z ,  ( F `  z ) >. } )
2321, 22syl 16 . . . . . . . . . . 11  |-  ( F  Fn  A  ->  ( F  |`  { z } )  C_  { <. z ,  ( F `  z ) >. } )
2423adantr 452 . . . . . . . . . 10  |-  ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  { z } )  C_  { <. z ,  ( F `  z ) >. } )
25 ssfi 7321 . . . . . . . . . 10  |-  ( ( { <. z ,  ( F `  z )
>. }  e.  Fin  /\  ( F  |`  { z } )  C_  { <. z ,  ( F `  z ) >. } )  ->  ( F  |`  { z } )  e.  Fin )
2620, 24, 25sylancr 645 . . . . . . . . 9  |-  ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  { z } )  e.  Fin )
27 unfi 7366 . . . . . . . . 9  |-  ( ( ( F  |`  y
)  e.  Fin  /\  ( F  |`  { z } )  e.  Fin )  ->  ( ( F  |`  y )  u.  ( F  |`  { z } ) )  e.  Fin )
2826, 27sylan2 461 . . . . . . . 8  |-  ( ( ( F  |`  y
)  e.  Fin  /\  ( F  Fn  A  /\  A  e.  Fin ) )  ->  (
( F  |`  y
)  u.  ( F  |`  { z } ) )  e.  Fin )
2919, 28syl5eqel 2519 . . . . . . 7  |-  ( ( ( F  |`  y
)  e.  Fin  /\  ( F  Fn  A  /\  A  e.  Fin ) )  ->  ( F  |`  ( y  u. 
{ z } ) )  e.  Fin )
3029expcom 425 . . . . . 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 7340 . . 3  |-  ( A  e.  Fin  ->  (
( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  A )  e.  Fin ) )
3433anabsi7 793 . 2  |-  ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  A )  e.  Fin )
352, 34eqeltrrd 2510 1  |-  ( ( F  Fn  A  /\  A  e.  Fin )  ->  F  e.  Fin )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    u. cun 3310    C_ wss 3312   (/)c0 3620   {csn 3806   <.cop 3809    |` cres 4872   Fun wfun 5440    Fn wfn 5441   ` cfv 5446   Fincfn 7101
This theorem is referenced by:  unirnffid  7390  mptfi  7398  seqf1olem2  11355  seqf1o  11356  iswrd  11721  wrdfin  11726  isstruct2  13470  xpsfrnel  13780  usgrafilem2  21418  sstotbnd2  26464  prdstotbnd  26484  stoweidlem59  27765  resfnfinfin  28061  hashfirdm  28133  hashfzdm  28134
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-fin 7105
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