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Theorem alephlim 7690
Description: Value of the aleph function at a limit ordinal. Definition 12(iii) of [Suppes] p. 91. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 13-Sep-2013.)
Assertion
Ref Expression
alephlim  |-  ( ( A  e.  V  /\  Lim  A )  ->  ( aleph `  A )  = 
U_ x  e.  A  ( aleph `  x )
)
Distinct variable group:    x, A
Allowed substitution hint:    V( x)

Proof of Theorem alephlim
StepHypRef Expression
1 rdglim2a 6442 . 2  |-  ( ( A  e.  V  /\  Lim  A )  ->  ( rec (har ,  om ) `  A )  =  U_ x  e.  A  ( rec (har ,  om ) `  x ) )
2 df-aleph 7569 . . 3  |-  aleph  =  rec (har ,  om )
32fveq1i 5487 . 2  |-  ( aleph `  A )  =  ( rec (har ,  om ) `  A )
42fveq1i 5487 . . . 4  |-  ( aleph `  x )  =  ( rec (har ,  om ) `  x )
54a1i 10 . . 3  |-  ( x  e.  A  ->  ( aleph `  x )  =  ( rec (har ,  om ) `  x ) )
65iuneq2i 3924 . 2  |-  U_ x  e.  A  ( aleph `  x )  =  U_ x  e.  A  ( rec (har ,  om ) `  x )
71, 3, 63eqtr4g 2341 1  |-  ( ( A  e.  V  /\  Lim  A )  ->  ( aleph `  A )  = 
U_ x  e.  A  ( aleph `  x )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1685   U_ciun 3906   Lim wlim 4392   omcom 4655   ` cfv 5221   reccrdg 6418  harchar 7266   alephcale 7565
This theorem is referenced by:  alephon  7692  alephcard  7693  alephordi  7697  cardaleph  7712  alephsing  7898  pwcfsdom  8201
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
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 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-recs 6384  df-rdg 6419  df-aleph 7569
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