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Theorem alephfplem3 8274
Description: Lemma for alephfp 8276. (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 5689 . . 3  |-  ( v  =  (/)  ->  ( H `
 v )  =  ( H `  (/) ) )
21eleq1d 2507 . 2  |-  ( v  =  (/)  ->  ( ( H `  v )  e.  ran  aleph  <->  ( H `  (/) )  e.  ran  aleph
) )
3 fveq2 5689 . . 3  |-  ( v  =  w  ->  ( H `  v )  =  ( H `  w ) )
43eleq1d 2507 . 2  |-  ( v  =  w  ->  (
( H `  v
)  e.  ran  aleph  <->  ( H `  w )  e.  ran  aleph
) )
5 fveq2 5689 . . 3  |-  ( v  =  suc  w  -> 
( H `  v
)  =  ( H `
 suc  w )
)
65eleq1d 2507 . 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 8272 . 2  |-  ( H `
 (/) )  e.  ran  aleph
9 alephfnon 8233 . . . 4  |-  aleph  Fn  On
10 alephsson 8268 . . . . 5  |-  ran  aleph  C_  On
1110sseli 3350 . . . 4  |-  ( ( H `  w )  e.  ran  aleph  ->  ( H `  w )  e.  On )
12 fnfvelrn 5838 . . . 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 8273 . . . 4  |-  ( w  e.  om  ->  ( H `  suc  w )  =  ( aleph `  ( H `  w )
) )
1514eleq1d 2507 . . 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 6503 1  |-  ( v  e.  om  ->  ( H `  v )  e.  ran  aleph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   (/)c0 3635   Oncon0 4717   suc csuc 4719   ran crn 4839    |` cres 4840    Fn wfn 5411   ` cfv 5416   omcom 6474   reccrdg 6863   alephcale 8104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-inf2 7845
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-se 4678  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-isom 5425  df-riota 6050  df-om 6475  df-recs 6830  df-rdg 6864  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-oi 7722  df-har 7771  df-card 8107  df-aleph 8108
This theorem is referenced by:  alephfplem4  8275  alephfp  8276
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