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Theorem alephfplem3 8812
 Description: Lemma for alephfp 8814. (Contributed by NM, 6-Nov-2004.)
Hypothesis
Ref Expression
alephfplem.1 𝐻 = (rec(ℵ, ω) ↾ ω)
Assertion
Ref Expression
alephfplem3 (𝑣 ∈ ω → (𝐻𝑣) ∈ ran ℵ)
Distinct variable group:   𝑣,𝐻

Proof of Theorem alephfplem3
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6103 . . 3 (𝑣 = ∅ → (𝐻𝑣) = (𝐻‘∅))
21eleq1d 2672 . 2 (𝑣 = ∅ → ((𝐻𝑣) ∈ ran ℵ ↔ (𝐻‘∅) ∈ ran ℵ))
3 fveq2 6103 . . 3 (𝑣 = 𝑤 → (𝐻𝑣) = (𝐻𝑤))
43eleq1d 2672 . 2 (𝑣 = 𝑤 → ((𝐻𝑣) ∈ ran ℵ ↔ (𝐻𝑤) ∈ ran ℵ))
5 fveq2 6103 . . 3 (𝑣 = suc 𝑤 → (𝐻𝑣) = (𝐻‘suc 𝑤))
65eleq1d 2672 . 2 (𝑣 = suc 𝑤 → ((𝐻𝑣) ∈ ran ℵ ↔ (𝐻‘suc 𝑤) ∈ ran ℵ))
7 alephfplem.1 . . 3 𝐻 = (rec(ℵ, ω) ↾ ω)
87alephfplem1 8810 . 2 (𝐻‘∅) ∈ ran ℵ
9 alephfnon 8771 . . . 4 ℵ Fn On
10 alephsson 8806 . . . . 5 ran ℵ ⊆ On
1110sseli 3564 . . . 4 ((𝐻𝑤) ∈ ran ℵ → (𝐻𝑤) ∈ On)
12 fnfvelrn 6264 . . . 4 ((ℵ Fn On ∧ (𝐻𝑤) ∈ On) → (ℵ‘(𝐻𝑤)) ∈ ran ℵ)
139, 11, 12sylancr 694 . . 3 ((𝐻𝑤) ∈ ran ℵ → (ℵ‘(𝐻𝑤)) ∈ ran ℵ)
147alephfplem2 8811 . . . 4 (𝑤 ∈ ω → (𝐻‘suc 𝑤) = (ℵ‘(𝐻𝑤)))
1514eleq1d 2672 . . 3 (𝑤 ∈ ω → ((𝐻‘suc 𝑤) ∈ ran ℵ ↔ (ℵ‘(𝐻𝑤)) ∈ ran ℵ))
1613, 15syl5ibr 235 . 2 (𝑤 ∈ ω → ((𝐻𝑤) ∈ ran ℵ → (𝐻‘suc 𝑤) ∈ ran ℵ))
172, 4, 6, 8, 16finds1 6987 1 (𝑣 ∈ ω → (𝐻𝑣) ∈ ran ℵ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ∅c0 3874  ran crn 5039   ↾ cres 5040  Oncon0 5640  suc csuc 5642   Fn wfn 5799  ‘cfv 5804  ωcom 6957  reccrdg 7392  ℵcale 8645 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-oi 8298  df-har 8346  df-card 8648  df-aleph 8649 This theorem is referenced by:  alephfplem4  8813  alephfp  8814
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