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Theorem alephlim 8834
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 7474 . 2 ((𝐴𝑉 ∧ Lim 𝐴) → (rec(har, ω)‘𝐴) = 𝑥𝐴 (rec(har, ω)‘𝑥))
2 df-aleph 8710 . . 3 ℵ = rec(har, ω)
32fveq1i 6149 . 2 (ℵ‘𝐴) = (rec(har, ω)‘𝐴)
42fveq1i 6149 . . . 4 (ℵ‘𝑥) = (rec(har, ω)‘𝑥)
54a1i 11 . . 3 (𝑥𝐴 → (ℵ‘𝑥) = (rec(har, ω)‘𝑥))
65iuneq2i 4505 . 2 𝑥𝐴 (ℵ‘𝑥) = 𝑥𝐴 (rec(har, ω)‘𝑥)
71, 3, 63eqtr4g 2680 1 ((𝐴𝑉 ∧ Lim 𝐴) → (ℵ‘𝐴) = 𝑥𝐴 (ℵ‘𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987   ciun 4485  Lim wlim 5683  cfv 5847  ωcom 7012  reccrdg 7450  harchar 8405  cale 8706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-aleph 8710
This theorem is referenced by:  alephon  8836  alephcard  8837  alephordi  8841  cardaleph  8856  alephsing  9042  pwcfsdom  9349
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