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

Theorem alephlim 9090
 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 7682 . 2 ((𝐴𝑉 ∧ Lim 𝐴) → (rec(har, ω)‘𝐴) = 𝑥𝐴 (rec(har, ω)‘𝑥))
2 df-aleph 8966 . . 3 ℵ = rec(har, ω)
32fveq1i 6333 . 2 (ℵ‘𝐴) = (rec(har, ω)‘𝐴)
42fveq1i 6333 . . . 4 (ℵ‘𝑥) = (rec(har, ω)‘𝑥)
54a1i 11 . . 3 (𝑥𝐴 → (ℵ‘𝑥) = (rec(har, ω)‘𝑥))
65iuneq2i 4673 . 2 𝑥𝐴 (ℵ‘𝑥) = 𝑥𝐴 (rec(har, ω)‘𝑥)
71, 3, 63eqtr4g 2830 1 ((𝐴𝑉 ∧ Lim 𝐴) → (ℵ‘𝐴) = 𝑥𝐴 (ℵ‘𝑥))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145  ∪ ciun 4654  Lim wlim 5867  ‘cfv 6031  ωcom 7212  reccrdg 7658  harchar 8617  ℵcale 8962 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-aleph 8966 This theorem is referenced by:  alephon  9092  alephcard  9093  alephordi  9097  cardaleph  9112  alephsing  9300  pwcfsdom  9607
 Copyright terms: Public domain W3C validator