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Theorem alephfplem4 8277
Description: Lemma for alephfp 8278. (Contributed by NM, 5-Nov-2004.)
Hypothesis
Ref Expression
alephfplem.1  |-  H  =  ( rec ( aleph ,  om )  |`  om )
Assertion
Ref Expression
alephfplem4  |-  U. ( H " om )  e. 
ran  aleph

Proof of Theorem alephfplem4
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 frfnom 6890 . . . . 5  |-  ( rec ( aleph ,  om )  |` 
om )  Fn  om
2 alephfplem.1 . . . . . 6  |-  H  =  ( rec ( aleph ,  om )  |`  om )
32fneq1i 5505 . . . . 5  |-  ( H  Fn  om  <->  ( rec ( aleph ,  om )  |` 
om )  Fn  om )
41, 3mpbir 209 . . . 4  |-  H  Fn  om
52alephfplem3 8276 . . . . 5  |-  ( z  e.  om  ->  ( H `  z )  e.  ran  aleph )
65rgen 2781 . . . 4  |-  A. z  e.  om  ( H `  z )  e.  ran  aleph
7 ffnfv 5869 . . . 4  |-  ( H : om --> ran  aleph  <->  ( H  Fn  om  /\  A. z  e.  om  ( H `  z )  e.  ran  aleph
) )
84, 6, 7mpbir2an 911 . . 3  |-  H : om
--> ran  aleph
9 ssun2 3520 . . 3  |-  ran  aleph  C_  ( om  u.  ran  aleph )
10 fss 5567 . . 3  |-  ( ( H : om --> ran  aleph  /\  ran  aleph  C_  ( om  u.  ran  aleph
) )  ->  H : om --> ( om  u.  ran  aleph ) )
118, 9, 10mp2an 672 . 2  |-  H : om
--> ( om  u.  ran  aleph
)
12 peano1 6495 . . 3  |-  (/)  e.  om
132alephfplem1 8274 . . 3  |-  ( H `
 (/) )  e.  ran  aleph
14 fveq2 5691 . . . . 5  |-  ( z  =  (/)  ->  ( H `
 z )  =  ( H `  (/) ) )
1514eleq1d 2509 . . . 4  |-  ( z  =  (/)  ->  ( ( H `  z )  e.  ran  aleph  <->  ( H `  (/) )  e.  ran  aleph
) )
1615rspcev 3073 . . 3  |-  ( (
(/)  e.  om  /\  ( H `  (/) )  e. 
ran  aleph )  ->  E. z  e.  om  ( H `  z )  e.  ran  aleph
)
1712, 13, 16mp2an 672 . 2  |-  E. z  e.  om  ( H `  z )  e.  ran  aleph
18 omex 7849 . . 3  |-  om  e.  _V
19 cardinfima 8267 . . 3  |-  ( om  e.  _V  ->  (
( H : om --> ( om  u.  ran  aleph )  /\  E. z  e.  om  ( H `  z )  e.  ran  aleph )  ->  U. ( H " om )  e. 
ran  aleph ) )
2018, 19ax-mp 5 . 2  |-  ( ( H : om --> ( om  u.  ran  aleph )  /\  E. z  e.  om  ( H `  z )  e.  ran  aleph )  ->  U. ( H " om )  e. 
ran  aleph )
2111, 17, 20mp2an 672 1  |-  U. ( H " om )  e. 
ran  aleph
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2715   E.wrex 2716   _Vcvv 2972    u. cun 3326    C_ wss 3328   (/)c0 3637   U.cuni 4091   ran crn 4841    |` cres 4842   "cima 4843    Fn wfn 5413   -->wf 5414   ` cfv 5418   omcom 6476   reccrdg 6865   alephcale 8106
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 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-om 6477  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-oi 7724  df-har 7773  df-card 8109  df-aleph 8110
This theorem is referenced by:  alephfp  8278  alephfp2  8279
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