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Theorem alephlim 9496
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 ((𝐴𝑉 ∧ Lim 𝐴) → (ℵ‘𝐴) = 𝑥𝐴 (ℵ‘𝑥))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem alephlim
StepHypRef Expression
1 rdglim2a 8072 . 2 ((𝐴𝑉 ∧ Lim 𝐴) → (rec(har, ω)‘𝐴) = 𝑥𝐴 (rec(har, ω)‘𝑥))
2 df-aleph 9372 . . 3 ℵ = rec(har, ω)
32fveq1i 6674 . 2 (ℵ‘𝐴) = (rec(har, ω)‘𝐴)
42fveq1i 6674 . . . 4 (ℵ‘𝑥) = (rec(har, ω)‘𝑥)
54a1i 11 . . 3 (𝑥𝐴 → (ℵ‘𝑥) = (rec(har, ω)‘𝑥))
65iuneq2i 4943 . 2 𝑥𝐴 (ℵ‘𝑥) = 𝑥𝐴 (rec(har, ω)‘𝑥)
71, 3, 63eqtr4g 2884 1 ((𝐴𝑉 ∧ Lim 𝐴) → (ℵ‘𝐴) = 𝑥𝐴 (ℵ‘𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1536  wcel 2113   ciun 4922  Lim wlim 6195  cfv 6358  ωcom 7583  reccrdg 8048  harchar 9023  cale 9368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-aleph 9372
This theorem is referenced by:  alephon  9498  alephcard  9499  alephordi  9503  cardaleph  9518  alephsing  9701  pwcfsdom  10008
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