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Theorem fnfi 7321
Description: A version of fnex 5901 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 5495 . . 3  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
21adantr 452 . 2  |-  ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  A )  =  F )
3 reseq2 5082 . . . . . 6  |-  ( x  =  (/)  ->  ( F  |`  x )  =  ( F  |`  (/) ) )
43eleq1d 2454 . . . . 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 5082 . . . . . 6  |-  ( x  =  y  ->  ( F  |`  x )  =  ( F  |`  y
) )
76eleq1d 2454 . . . . 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 5082 . . . . . 6  |-  ( x  =  ( y  u. 
{ z } )  ->  ( F  |`  x )  =  ( F  |`  ( y  u.  { z } ) ) )
109eleq1d 2454 . . . . 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 5082 . . . . . 6  |-  ( x  =  A  ->  ( F  |`  x )  =  ( F  |`  A ) )
1312eleq1d 2454 . . . . 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 5091 . . . . . 6  |-  ( F  |`  (/) )  =  (/)
16 0fin 7273 . . . . . 6  |-  (/)  e.  Fin
1715, 16eqeltri 2458 . . . . 5  |-  ( F  |`  (/) )  e.  Fin
1817a1i 11 . . . 4  |-  ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  (/) )  e. 
Fin )
19 resundi 5101 . . . . . . . 8  |-  ( F  |`  ( y  u.  {
z } ) )  =  ( ( F  |`  y )  u.  ( F  |`  { z } ) )
20 snfi 7124 . . . . . . . . . 10  |-  { <. z ,  ( F `  z ) >. }  e.  Fin
21 fnfun 5483 . . . . . . . . . . . 12  |-  ( F  Fn  A  ->  Fun  F )
22 funressn 5859 . . . . . . . . . . . 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 7266 . . . . . . . . . 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 7311 . . . . . . . . 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 2472 . . . . . . 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 7285 . . 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 2463 1  |-  ( ( F  Fn  A  /\  A  e.  Fin )  ->  F  e.  Fin )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    u. cun 3262    C_ wss 3264   (/)c0 3572   {csn 3758   <.cop 3761    |` cres 4821   Fun wfun 5389    Fn wfn 5390   ` cfv 5395   Fincfn 7046
This theorem is referenced by:  unirnffid  7334  mptfi  7342  seqf1olem2  11291  seqf1o  11292  iswrd  11657  wrdfin  11662  isstruct2  13406  xpsfrnel  13716  usgrafilem2  21293  sstotbnd2  26175  prdstotbnd  26195  stoweidlem59  27477
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-er 6842  df-en 7047  df-fin 7050
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