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Theorem alephfplem3 7749
Description: Lemma for alephfp 7751. (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 5541 . . 3  |-  ( v  =  (/)  ->  ( H `
 v )  =  ( H `  (/) ) )
21eleq1d 2362 . 2  |-  ( v  =  (/)  ->  ( ( H `  v )  e.  ran  aleph  <->  ( H `  (/) )  e.  ran  aleph
) )
3 fveq2 5541 . . 3  |-  ( v  =  w  ->  ( H `  v )  =  ( H `  w ) )
43eleq1d 2362 . 2  |-  ( v  =  w  ->  (
( H `  v
)  e.  ran  aleph  <->  ( H `  w )  e.  ran  aleph
) )
5 fveq2 5541 . . 3  |-  ( v  =  suc  w  -> 
( H `  v
)  =  ( H `
 suc  w )
)
65eleq1d 2362 . 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 7747 . 2  |-  ( H `
 (/) )  e.  ran  aleph
9 alephfnon 7708 . . . 4  |-  aleph  Fn  On
10 alephsson 7743 . . . . 5  |-  ran  aleph  C_  On
1110sseli 3189 . . . 4  |-  ( ( H `  w )  e.  ran  aleph  ->  ( H `  w )  e.  On )
12 fnfvelrn 5678 . . . 4  |-  ( (
aleph  Fn  On  /\  ( H `  w )  e.  On )  ->  ( aleph `  ( H `  w ) )  e. 
ran  aleph )
139, 11, 12sylancr 644 . . 3  |-  ( ( H `  w )  e.  ran  aleph  ->  ( aleph `  ( H `  w ) )  e. 
ran  aleph )
147alephfplem2 7748 . . . 4  |-  ( w  e.  om  ->  ( H `  suc  w )  =  ( aleph `  ( H `  w )
) )
1514eleq1d 2362 . . 3  |-  ( w  e.  om  ->  (
( H `  suc  w )  e.  ran  aleph  <->  (
aleph `  ( H `  w ) )  e. 
ran  aleph ) )
1613, 15syl5ibr 212 . 2  |-  ( w  e.  om  ->  (
( H `  w
)  e.  ran  aleph  ->  ( H `  suc  w )  e.  ran  aleph ) )
172, 4, 6, 8, 16finds1 4701 1  |-  ( v  e.  om  ->  ( H `  v )  e.  ran  aleph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   (/)c0 3468   Oncon0 4408   suc csuc 4410   omcom 4672   ran crn 4706    |` cres 4707    Fn wfn 5266   ` cfv 5271   reccrdg 6438   alephcale 7585
This theorem is referenced by:  alephfplem4  7750  alephfp  7751
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-har 7288  df-card 7588  df-aleph 7589
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