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Theorem alephfplem3 7922
Description: Lemma for alephfp 7924. (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 5670 . . 3  |-  ( v  =  (/)  ->  ( H `
 v )  =  ( H `  (/) ) )
21eleq1d 2455 . 2  |-  ( v  =  (/)  ->  ( ( H `  v )  e.  ran  aleph  <->  ( H `  (/) )  e.  ran  aleph
) )
3 fveq2 5670 . . 3  |-  ( v  =  w  ->  ( H `  v )  =  ( H `  w ) )
43eleq1d 2455 . 2  |-  ( v  =  w  ->  (
( H `  v
)  e.  ran  aleph  <->  ( H `  w )  e.  ran  aleph
) )
5 fveq2 5670 . . 3  |-  ( v  =  suc  w  -> 
( H `  v
)  =  ( H `
 suc  w )
)
65eleq1d 2455 . 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 7920 . 2  |-  ( H `
 (/) )  e.  ran  aleph
9 alephfnon 7881 . . . 4  |-  aleph  Fn  On
10 alephsson 7916 . . . . 5  |-  ran  aleph  C_  On
1110sseli 3289 . . . 4  |-  ( ( H `  w )  e.  ran  aleph  ->  ( H `  w )  e.  On )
12 fnfvelrn 5808 . . . 4  |-  ( (
aleph  Fn  On  /\  ( H `  w )  e.  On )  ->  ( aleph `  ( H `  w ) )  e. 
ran  aleph )
139, 11, 12sylancr 645 . . 3  |-  ( ( H `  w )  e.  ran  aleph  ->  ( aleph `  ( H `  w ) )  e. 
ran  aleph )
147alephfplem2 7921 . . . 4  |-  ( w  e.  om  ->  ( H `  suc  w )  =  ( aleph `  ( H `  w )
) )
1514eleq1d 2455 . . 3  |-  ( w  e.  om  ->  (
( H `  suc  w )  e.  ran  aleph  <->  (
aleph `  ( H `  w ) )  e. 
ran  aleph ) )
1613, 15syl5ibr 213 . 2  |-  ( w  e.  om  ->  (
( H `  w
)  e.  ran  aleph  ->  ( H `  suc  w )  e.  ran  aleph ) )
172, 4, 6, 8, 16finds1 4816 1  |-  ( v  e.  om  ->  ( H `  v )  e.  ran  aleph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   (/)c0 3573   Oncon0 4524   suc csuc 4526   omcom 4787   ran crn 4821    |` cres 4822    Fn wfn 5391   ` cfv 5396   reccrdg 6605   alephcale 7758
This theorem is referenced by:  alephfplem4  7923  alephfp  7924
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 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-inf2 7531
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-se 4485  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-isom 5405  df-riota 6487  df-recs 6571  df-rdg 6606  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-oi 7414  df-har 7461  df-card 7761  df-aleph 7762
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