MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  alephlim Unicode version

Theorem alephlim 7904
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 6650 . 2  |-  ( ( A  e.  V  /\  Lim  A )  ->  ( rec (har ,  om ) `  A )  =  U_ x  e.  A  ( rec (har ,  om ) `  x ) )
2 df-aleph 7783 . . 3  |-  aleph  =  rec (har ,  om )
32fveq1i 5688 . 2  |-  ( aleph `  A )  =  ( rec (har ,  om ) `  A )
42fveq1i 5688 . . . 4  |-  ( aleph `  x )  =  ( rec (har ,  om ) `  x )
54a1i 11 . . 3  |-  ( x  e.  A  ->  ( aleph `  x )  =  ( rec (har ,  om ) `  x ) )
65iuneq2i 4071 . 2  |-  U_ x  e.  A  ( aleph `  x )  =  U_ x  e.  A  ( rec (har ,  om ) `  x )
71, 3, 63eqtr4g 2461 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 359    = wceq 1649    e. wcel 1721   U_ciun 4053   Lim wlim 4542   omcom 4804   ` cfv 5413   reccrdg 6626  harchar 7480   alephcale 7779
This theorem is referenced by:  alephon  7906  alephcard  7907  alephordi  7911  cardaleph  7926  alephsing  8112  pwcfsdom  8414
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-recs 6592  df-rdg 6627  df-aleph 7783
  Copyright terms: Public domain W3C validator