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Theorem fnfi 7816
Description: A version of fnex 6140 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 5696 . . 3  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
21adantr 465 . 2  |-  ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  A )  =  F )
3 reseq2 5278 . . . . . 6  |-  ( x  =  (/)  ->  ( F  |`  x )  =  ( F  |`  (/) ) )
43eleq1d 2526 . . . . 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 5278 . . . . . 6  |-  ( x  =  y  ->  ( F  |`  x )  =  ( F  |`  y
) )
76eleq1d 2526 . . . . 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 5278 . . . . . 6  |-  ( x  =  ( y  u. 
{ z } )  ->  ( F  |`  x )  =  ( F  |`  ( y  u.  { z } ) ) )
109eleq1d 2526 . . . . 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 5278 . . . . . 6  |-  ( x  =  A  ->  ( F  |`  x )  =  ( F  |`  A ) )
1312eleq1d 2526 . . . . 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 5288 . . . . . 6  |-  ( F  |`  (/) )  =  (/)
16 0fin 7766 . . . . . 6  |-  (/)  e.  Fin
1715, 16eqeltri 2541 . . . . 5  |-  ( F  |`  (/) )  e.  Fin
1817a1i 11 . . . 4  |-  ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  (/) )  e. 
Fin )
19 resundi 5297 . . . . . . . 8  |-  ( F  |`  ( y  u.  {
z } ) )  =  ( ( F  |`  y )  u.  ( F  |`  { z } ) )
20 snfi 7615 . . . . . . . . . 10  |-  { <. z ,  ( F `  z ) >. }  e.  Fin
21 fnfun 5684 . . . . . . . . . . . 12  |-  ( F  Fn  A  ->  Fun  F )
22 funressn 6085 . . . . . . . . . . . 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 7759 . . . . . . . . . 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 7805 . . . . . . . . 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 2549 . . . . . . 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 7778 . . 3  |-  ( A  e.  Fin  ->  (
( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  A )  e.  Fin ) )
3433anabsi7 819 . 2  |-  ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  A )  e.  Fin )
352, 34eqeltrrd 2546 1  |-  ( ( F  Fn  A  /\  A  e.  Fin )  ->  F  e.  Fin )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819    u. cun 3469    C_ wss 3471   (/)c0 3793   {csn 4032   <.cop 4038    |` cres 5010   Fun wfun 5588    Fn wfn 5589   ` cfv 5594   Fincfn 7535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-fin 7539
This theorem is referenced by:  unirnffid  7830  mptfi  7837  seqf1olem2  12149  seqf1o  12150  hashfzdm  12501  hashfirdm  12503  iswrdOLD  12554  wrdfin  12567  isstruct2  14652  xpsfrnel  14979  usgrafilem2  24538  cmpcref  28006  ftc1anclem3  30254  sstotbnd2  30432  prdstotbnd  30452  ffi  31611  stoweidlem59  32002  fourierdlem42  32092  fourierdlem54  32104  resfnfinfin  32532  fundmfibi  32533
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