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Theorem alephfplem3 8504
Description: Lemma for alephfp 8506. (Contributed by NM, 6-Nov-2004.)
Hypothesis
Ref Expression
alephfplem.1  |-  H  =  ( rec ( aleph ,  om )  |`  om )
Assertion
Ref Expression
alephfplem3  |-  ( v  e.  om  ->  ( H `  v )  e.  ran  aleph )
Distinct variable group:    v, H

Proof of Theorem alephfplem3
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 fveq2 5872 . . 3  |-  ( v  =  (/)  ->  ( H `
 v )  =  ( H `  (/) ) )
21eleq1d 2526 . 2  |-  ( v  =  (/)  ->  ( ( H `  v )  e.  ran  aleph  <->  ( H `  (/) )  e.  ran  aleph
) )
3 fveq2 5872 . . 3  |-  ( v  =  w  ->  ( H `  v )  =  ( H `  w ) )
43eleq1d 2526 . 2  |-  ( v  =  w  ->  (
( H `  v
)  e.  ran  aleph  <->  ( H `  w )  e.  ran  aleph
) )
5 fveq2 5872 . . 3  |-  ( v  =  suc  w  -> 
( H `  v
)  =  ( H `
 suc  w )
)
65eleq1d 2526 . 2  |-  ( v  =  suc  w  -> 
( ( H `  v )  e.  ran  aleph  <->  ( H `  suc  w
)  e.  ran  aleph ) )
7 alephfplem.1 . . 3  |-  H  =  ( rec ( aleph ,  om )  |`  om )
87alephfplem1 8502 . 2  |-  ( H `
 (/) )  e.  ran  aleph
9 alephfnon 8463 . . . 4  |-  aleph  Fn  On
10 alephsson 8498 . . . . 5  |-  ran  aleph  C_  On
1110sseli 3495 . . . 4  |-  ( ( H `  w )  e.  ran  aleph  ->  ( H `  w )  e.  On )
12 fnfvelrn 6029 . . . 4  |-  ( (
aleph  Fn  On  /\  ( H `  w )  e.  On )  ->  ( aleph `  ( H `  w ) )  e. 
ran  aleph )
139, 11, 12sylancr 663 . . 3  |-  ( ( H `  w )  e.  ran  aleph  ->  ( aleph `  ( H `  w ) )  e. 
ran  aleph )
147alephfplem2 8503 . . . 4  |-  ( w  e.  om  ->  ( H `  suc  w )  =  ( aleph `  ( H `  w )
) )
1514eleq1d 2526 . . 3  |-  ( w  e.  om  ->  (
( H `  suc  w )  e.  ran  aleph  <->  (
aleph `  ( H `  w ) )  e. 
ran  aleph ) )
1613, 15syl5ibr 221 . 2  |-  ( w  e.  om  ->  (
( H `  w
)  e.  ran  aleph  ->  ( H `  suc  w )  e.  ran  aleph ) )
172, 4, 6, 8, 16finds1 6728 1  |-  ( v  e.  om  ->  ( H `  v )  e.  ran  aleph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395    e. wcel 1819   (/)c0 3793   Oncon0 4887   suc csuc 4889   ran crn 5009    |` cres 5010    Fn wfn 5589   ` cfv 5594   omcom 6699   reccrdg 7093   alephcale 8334
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-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075
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-rmo 2815  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-se 4848  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-isom 5603  df-riota 6258  df-om 6700  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-oi 7953  df-har 8002  df-card 8337  df-aleph 8338
This theorem is referenced by:  alephfplem4  8505  alephfp  8506
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