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Theorem r1fin 7535
Description: The first  om levels of the cumulative hierarchy are all finite. (Contributed by Mario Carneiro, 15-May-2013.)
Assertion
Ref Expression
r1fin  |-  ( A  e.  om  ->  ( R1 `  A )  e. 
Fin )

Proof of Theorem r1fin
Dummy variables  n  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5608 . . 3  |-  ( n  =  (/)  ->  ( R1
`  n )  =  ( R1 `  (/) ) )
21eleq1d 2424 . 2  |-  ( n  =  (/)  ->  ( ( R1 `  n )  e.  Fin  <->  ( R1 `  (/) )  e.  Fin ) )
3 fveq2 5608 . . 3  |-  ( n  =  m  ->  ( R1 `  n )  =  ( R1 `  m
) )
43eleq1d 2424 . 2  |-  ( n  =  m  ->  (
( R1 `  n
)  e.  Fin  <->  ( R1 `  m )  e.  Fin ) )
5 fveq2 5608 . . 3  |-  ( n  =  suc  m  -> 
( R1 `  n
)  =  ( R1
`  suc  m )
)
65eleq1d 2424 . 2  |-  ( n  =  suc  m  -> 
( ( R1 `  n )  e.  Fin  <->  ( R1 `  suc  m )  e.  Fin ) )
7 fveq2 5608 . . 3  |-  ( n  =  A  ->  ( R1 `  n )  =  ( R1 `  A
) )
87eleq1d 2424 . 2  |-  ( n  =  A  ->  (
( R1 `  n
)  e.  Fin  <->  ( R1 `  A )  e.  Fin ) )
9 r10 7530 . . 3  |-  ( R1
`  (/) )  =  (/)
10 0fin 7177 . . 3  |-  (/)  e.  Fin
119, 10eqeltri 2428 . 2  |-  ( R1
`  (/) )  e.  Fin
12 r1funlim 7528 . . . . . . . . 9  |-  ( Fun 
R1  /\  Lim  dom  R1 )
1312simpri 448 . . . . . . . 8  |-  Lim  dom  R1
14 limomss 4743 . . . . . . . 8  |-  ( Lim 
dom  R1  ->  om  C_  dom  R1 )
1513, 14ax-mp 8 . . . . . . 7  |-  om  C_  dom  R1
1615sseli 3252 . . . . . 6  |-  ( m  e.  om  ->  m  e.  dom  R1 )
17 r1sucg 7531 . . . . . 6  |-  ( m  e.  dom  R1  ->  ( R1 `  suc  m
)  =  ~P ( R1 `  m ) )
1816, 17syl 15 . . . . 5  |-  ( m  e.  om  ->  ( R1 `  suc  m )  =  ~P ( R1
`  m ) )
1918eleq1d 2424 . . . 4  |-  ( m  e.  om  ->  (
( R1 `  suc  m )  e.  Fin  <->  ~P ( R1 `  m )  e.  Fin ) )
20 pwfi 7241 . . . 4  |-  ( ( R1 `  m )  e.  Fin  <->  ~P ( R1 `  m )  e. 
Fin )
2119, 20syl6rbbr 255 . . 3  |-  ( m  e.  om  ->  (
( R1 `  m
)  e.  Fin  <->  ( R1 ` 
suc  m )  e. 
Fin ) )
2221biimpd 198 . 2  |-  ( m  e.  om  ->  (
( R1 `  m
)  e.  Fin  ->  ( R1 `  suc  m
)  e.  Fin )
)
232, 4, 6, 8, 11, 22finds 4764 1  |-  ( A  e.  om  ->  ( R1 `  A )  e. 
Fin )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1642    e. wcel 1710    C_ wss 3228   (/)c0 3531   ~Pcpw 3701   Lim wlim 4475   suc csuc 4476   omcom 4738   dom cdm 4771   Fun wfun 5331   ` cfv 5337   Fincfn 6951   R1cr1 7524
This theorem is referenced by:  ackbij2lem2  7956  ackbij2  7959
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-recs 6475  df-rdg 6510  df-1o 6566  df-2o 6567  df-oadd 6570  df-er 6747  df-map 6862  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-r1 7526
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