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Theorem alephfplem3 9129
Description: Lemma for alephfp 9131. (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 6332 . . 3 (𝑣 = ∅ → (𝐻𝑣) = (𝐻‘∅))
21eleq1d 2835 . 2 (𝑣 = ∅ → ((𝐻𝑣) ∈ ran ℵ ↔ (𝐻‘∅) ∈ ran ℵ))
3 fveq2 6332 . . 3 (𝑣 = 𝑤 → (𝐻𝑣) = (𝐻𝑤))
43eleq1d 2835 . 2 (𝑣 = 𝑤 → ((𝐻𝑣) ∈ ran ℵ ↔ (𝐻𝑤) ∈ ran ℵ))
5 fveq2 6332 . . 3 (𝑣 = suc 𝑤 → (𝐻𝑣) = (𝐻‘suc 𝑤))
65eleq1d 2835 . 2 (𝑣 = suc 𝑤 → ((𝐻𝑣) ∈ ran ℵ ↔ (𝐻‘suc 𝑤) ∈ ran ℵ))
7 alephfplem.1 . . 3 𝐻 = (rec(ℵ, ω) ↾ ω)
87alephfplem1 9127 . 2 (𝐻‘∅) ∈ ran ℵ
9 alephfnon 9088 . . . 4 ℵ Fn On
10 alephsson 9123 . . . . 5 ran ℵ ⊆ On
1110sseli 3748 . . . 4 ((𝐻𝑤) ∈ ran ℵ → (𝐻𝑤) ∈ On)
12 fnfvelrn 6499 . . . 4 ((ℵ Fn On ∧ (𝐻𝑤) ∈ On) → (ℵ‘(𝐻𝑤)) ∈ ran ℵ)
139, 11, 12sylancr 575 . . 3 ((𝐻𝑤) ∈ ran ℵ → (ℵ‘(𝐻𝑤)) ∈ ran ℵ)
147alephfplem2 9128 . . . 4 (𝑤 ∈ ω → (𝐻‘suc 𝑤) = (ℵ‘(𝐻𝑤)))
1514eleq1d 2835 . . 3 (𝑤 ∈ ω → ((𝐻‘suc 𝑤) ∈ ran ℵ ↔ (ℵ‘(𝐻𝑤)) ∈ ran ℵ))
1613, 15syl5ibr 236 . 2 (𝑤 ∈ ω → ((𝐻𝑤) ∈ ran ℵ → (𝐻‘suc 𝑤) ∈ ran ℵ))
172, 4, 6, 8, 16finds1 7242 1 (𝑣 ∈ ω → (𝐻𝑣) ∈ ran ℵ)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  c0 4063  ran crn 5250  cres 5251  Oncon0 5866  suc csuc 5868   Fn wfn 6026  cfv 6031  ωcom 7212  reccrdg 7658  cale 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  ax-inf2 8702
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-rmo 3069  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-int 4612  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-se 5209  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-isom 6040  df-riota 6754  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-oi 8571  df-har 8619  df-card 8965  df-aleph 8966
This theorem is referenced by:  alephfplem4  9130  alephfp  9131
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