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Theorem alephlim 7627
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 6379 . 2  |-  ( ( A  e.  V  /\  Lim  A )  ->  ( rec (har ,  om ) `  A )  =  U_ x  e.  A  ( rec (har ,  om ) `  x ) )
2 df-aleph 7506 . . 3  |-  aleph  =  rec (har ,  om )
32fveq1i 5424 . 2  |-  ( aleph `  A )  =  ( rec (har ,  om ) `  A )
42fveq1i 5424 . . . 4  |-  ( aleph `  x )  =  ( rec (har ,  om ) `  x )
54a1i 12 . . 3  |-  ( x  e.  A  ->  ( aleph `  x )  =  ( rec (har ,  om ) `  x ) )
65iuneq2i 3864 . 2  |-  U_ x  e.  A  ( aleph `  x )  =  U_ x  e.  A  ( rec (har ,  om ) `  x )
71, 3, 63eqtr4g 2313 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 3846   Lim wlim 4330   omcom 4593   ` cfv 4638   reccrdg 6355  harchar 7203   alephcale 7502
This theorem is referenced by:  alephon  7629  alephcard  7630  alephordi  7634  cardaleph  7649  alephsing  7835  pwcfsdom  8138
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 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pr 4152  ax-un 4449
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-recs 6321  df-rdg 6356  df-aleph 7506
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