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Theorem fnfi 7134
Description: A version of fnex 5741 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 5353 . . 3  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
21adantr 451 . 2  |-  ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  A )  =  F )
3 reseq2 4950 . . . . . 6  |-  ( x  =  (/)  ->  ( F  |`  x )  =  ( F  |`  (/) ) )
43eleq1d 2349 . . . . 5  |-  ( x  =  (/)  ->  ( ( F  |`  x )  e.  Fin  <->  ( F  |`  (/) )  e.  Fin )
)
54imbi2d 307 . . . 4  |-  ( x  =  (/)  ->  ( ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  x )  e.  Fin ) 
<->  ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  (/) )  e.  Fin )
) )
6 reseq2 4950 . . . . . 6  |-  ( x  =  y  ->  ( F  |`  x )  =  ( F  |`  y
) )
76eleq1d 2349 . . . . 5  |-  ( x  =  y  ->  (
( F  |`  x
)  e.  Fin  <->  ( F  |`  y )  e.  Fin ) )
87imbi2d 307 . . . 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 4950 . . . . . 6  |-  ( x  =  ( y  u. 
{ z } )  ->  ( F  |`  x )  =  ( F  |`  ( y  u.  { z } ) ) )
109eleq1d 2349 . . . . 5  |-  ( x  =  ( y  u. 
{ z } )  ->  ( ( F  |`  x )  e.  Fin  <->  ( F  |`  ( y  u. 
{ z } ) )  e.  Fin )
)
1110imbi2d 307 . . . 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 4950 . . . . . 6  |-  ( x  =  A  ->  ( F  |`  x )  =  ( F  |`  A ) )
1312eleq1d 2349 . . . . 5  |-  ( x  =  A  ->  (
( F  |`  x
)  e.  Fin  <->  ( F  |`  A )  e.  Fin ) )
1413imbi2d 307 . . . 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 4959 . . . . . 6  |-  ( F  |`  (/) )  =  (/)
16 0fin 7087 . . . . . 6  |-  (/)  e.  Fin
1715, 16eqeltri 2353 . . . . 5  |-  ( F  |`  (/) )  e.  Fin
1817a1i 10 . . . 4  |-  ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  (/) )  e. 
Fin )
19 resundi 4969 . . . . . . . 8  |-  ( F  |`  ( y  u.  {
z } ) )  =  ( ( F  |`  y )  u.  ( F  |`  { z } ) )
20 snfi 6941 . . . . . . . . . 10  |-  { <. z ,  ( F `  z ) >. }  e.  Fin
21 fnfun 5341 . . . . . . . . . . . 12  |-  ( F  Fn  A  ->  Fun  F )
22 funressn 5706 . . . . . . . . . . . 12  |-  ( Fun 
F  ->  ( F  |` 
{ z } ) 
C_  { <. z ,  ( F `  z ) >. } )
2321, 22syl 15 . . . . . . . . . . 11  |-  ( F  Fn  A  ->  ( F  |`  { z } )  C_  { <. z ,  ( F `  z ) >. } )
2423adantr 451 . . . . . . . . . 10  |-  ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  { z } )  C_  { <. z ,  ( F `  z ) >. } )
25 ssfi 7083 . . . . . . . . . 10  |-  ( ( { <. z ,  ( F `  z )
>. }  e.  Fin  /\  ( F  |`  { z } )  C_  { <. z ,  ( F `  z ) >. } )  ->  ( F  |`  { z } )  e.  Fin )
2620, 24, 25sylancr 644 . . . . . . . . 9  |-  ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  { z } )  e.  Fin )
27 unfi 7124 . . . . . . . . 9  |-  ( ( ( F  |`  y
)  e.  Fin  /\  ( F  |`  { z } )  e.  Fin )  ->  ( ( F  |`  y )  u.  ( F  |`  { z } ) )  e.  Fin )
2826, 27sylan2 460 . . . . . . . 8  |-  ( ( ( F  |`  y
)  e.  Fin  /\  ( F  Fn  A  /\  A  e.  Fin ) )  ->  (
( F  |`  y
)  u.  ( F  |`  { z } ) )  e.  Fin )
2919, 28syl5eqel 2367 . . . . . . 7  |-  ( ( ( F  |`  y
)  e.  Fin  /\  ( F  Fn  A  /\  A  e.  Fin ) )  ->  ( F  |`  ( y  u. 
{ z } ) )  e.  Fin )
3029expcom 424 . . . . . 6  |-  ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( ( F  |`  y )  e.  Fin  ->  ( F  |`  (
y  u.  { z } ) )  e. 
Fin ) )
3130a2i 12 . . . . 5  |-  ( ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  y )  e.  Fin )  ->  ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  ( y  u. 
{ z } ) )  e.  Fin )
)
3231a1i 10 . . . 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 7098 . . 3  |-  ( A  e.  Fin  ->  (
( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  A )  e.  Fin ) )
3433anabsi7 792 . 2  |-  ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( F  |`  A )  e.  Fin )
352, 34eqeltrrd 2358 1  |-  ( ( F  Fn  A  /\  A  e.  Fin )  ->  F  e.  Fin )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    u. cun 3150    C_ wss 3152   (/)c0 3455   {csn 3640   <.cop 3643    |` cres 4691   Fun wfun 5249    Fn wfn 5250   ` cfv 5255   Fincfn 6863
This theorem is referenced by:  unirnffid  7147  mptfi  7155  seqf1olem2  11086  seqf1o  11087  iswrd  11415  wrdfin  11420  isstruct2  13157  xpsfrnel  13465  sstotbnd2  26498  prdstotbnd  26518  stoweidlem59  27808
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-fin 6867
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