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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  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  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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