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

Theorem alephlim 8351
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 7002 . 2  |-  ( ( A  e.  V  /\  Lim  A )  ->  ( rec (har ,  om ) `  A )  =  U_ x  e.  A  ( rec (har ,  om ) `  x ) )
2 df-aleph 8224 . . 3  |-  aleph  =  rec (har ,  om )
32fveq1i 5803 . 2  |-  ( aleph `  A )  =  ( rec (har ,  om ) `  A )
42fveq1i 5803 . . . 4  |-  ( aleph `  x )  =  ( rec (har ,  om ) `  x )
54a1i 11 . . 3  |-  ( x  e.  A  ->  ( aleph `  x )  =  ( rec (har ,  om ) `  x ) )
65iuneq2i 4300 . 2  |-  U_ x  e.  A  ( aleph `  x )  =  U_ x  e.  A  ( rec (har ,  om ) `  x )
71, 3, 63eqtr4g 2520 1  |-  ( ( A  e.  V  /\  Lim  A )  ->  ( aleph `  A )  = 
U_ x  e.  A  ( aleph `  x )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   U_ciun 4282   Lim wlim 4831   ` cfv 5529   omcom 6589   reccrdg 6978  harchar 7885   alephcale 8220
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-recs 6945  df-rdg 6979  df-aleph 8224
This theorem is referenced by:  alephon  8353  alephcard  8354  alephordi  8358  cardaleph  8373  alephsing  8559  pwcfsdom  8861
  Copyright terms: Public domain W3C validator