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Theorem alephlim 7578
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 6332 . 2  |-  ( ( A  e.  V  /\  Lim  A )  ->  ( rec (har ,  om ) `  A )  =  U_ x  e.  A  ( rec (har ,  om ) `  x ) )
2 df-aleph 7457 . . 3  |-  aleph  =  rec (har ,  om )
32fveq1i 5378 . 2  |-  ( aleph `  A )  =  ( rec (har ,  om ) `  A )
42fveq1i 5378 . . . 4  |-  ( aleph `  x )  =  ( rec (har ,  om ) `  x )
54a1i 12 . . 3  |-  ( x  e.  A  ->  ( aleph `  x )  =  ( rec (har ,  om ) `  x ) )
65iuneq2i 3821 . 2  |-  U_ x  e.  A  ( aleph `  x )  =  U_ x  e.  A  ( rec (har ,  om ) `  x )
71, 3, 63eqtr4g 2310 1  |-  ( ( A  e.  V  /\  Lim  A )  ->  ( aleph `  A )  = 
U_ x  e.  A  ( aleph `  x )
)
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   U_ciun 3803   Lim wlim 4286   omcom 4547   ` cfv 4592   reccrdg 6308  harchar 7154   alephcale 7453
This theorem is referenced by:  alephon  7580  alephcard  7581  alephordi  7585  cardaleph  7600  alephsing  7786  pwcfsdom  8085
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-recs 6274  df-rdg 6309  df-aleph 7457
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