Theorem alephfplem3 10097

Description: Lemma for alephfp 10099. (Contributed by NM, 6-Nov-2004.)

Hypothesis

Ref	Expression
alephfplem.1	⊢ 𝐻 = (rec(ℵ, ω) ↾ ω)

Assertion

Ref	Expression
alephfplem3	⊢ (𝑣 ∈ ω → (𝐻‘𝑣) ∈ ran ℵ)

Distinct variable group:   𝑣,𝐻

Proof of Theorem alephfplem3

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6888	. . 3 ⊢ (𝑣 = ∅ → (𝐻‘𝑣) = (𝐻‘∅))
2	1	eleq1d 2818	. 2 ⊢ (𝑣 = ∅ → ((𝐻‘𝑣) ∈ ran ℵ ↔ (𝐻‘∅) ∈ ran ℵ))
3		fveq2 6888	. . 3 ⊢ (𝑣 = 𝑤 → (𝐻‘𝑣) = (𝐻‘𝑤))
4	3	eleq1d 2818	. 2 ⊢ (𝑣 = 𝑤 → ((𝐻‘𝑣) ∈ ran ℵ ↔ (𝐻‘𝑤) ∈ ran ℵ))
5		fveq2 6888	. . 3 ⊢ (𝑣 = suc 𝑤 → (𝐻‘𝑣) = (𝐻‘suc 𝑤))
6	5	eleq1d 2818	. 2 ⊢ (𝑣 = suc 𝑤 → ((𝐻‘𝑣) ∈ ran ℵ ↔ (𝐻‘suc 𝑤) ∈ ran ℵ))
7		alephfplem.1	. . 3 ⊢ 𝐻 = (rec(ℵ, ω) ↾ ω)
8	7	alephfplem1 10095	. 2 ⊢ (𝐻‘∅) ∈ ran ℵ
9		alephfnon 10056	. . . 4 ⊢ ℵ Fn On
10		alephsson 10091	. . . . 5 ⊢ ran ℵ ⊆ On
11	10	sseli 3977	. . . 4 ⊢ ((𝐻‘𝑤) ∈ ran ℵ → (𝐻‘𝑤) ∈ On)
12		fnfvelrn 7079	. . . 4 ⊢ ((ℵ Fn On ∧ (𝐻‘𝑤) ∈ On) → (ℵ‘(𝐻‘𝑤)) ∈ ran ℵ)
13	9, 11, 12	sylancr 587	. . 3 ⊢ ((𝐻‘𝑤) ∈ ran ℵ → (ℵ‘(𝐻‘𝑤)) ∈ ran ℵ)
14	7	alephfplem2 10096	. . . 4 ⊢ (𝑤 ∈ ω → (𝐻‘suc 𝑤) = (ℵ‘(𝐻‘𝑤)))
15	14	eleq1d 2818	. . 3 ⊢ (𝑤 ∈ ω → ((𝐻‘suc 𝑤) ∈ ran ℵ ↔ (ℵ‘(𝐻‘𝑤)) ∈ ran ℵ))
16	13, 15	imbitrrid 245	. 2 ⊢ (𝑤 ∈ ω → ((𝐻‘𝑤) ∈ ran ℵ → (𝐻‘suc 𝑤) ∈ ran ℵ))
17	2, 4, 6, 8, 16	finds1 7888	1 ⊢ (𝑣 ∈ ω → (𝐻‘𝑣) ∈ ran ℵ)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2106  ∅c0 4321  ran crn 5676   ↾ cres 5677  Oncon0 6361  suc csuc 6363   Fn wfn 6535  ‘cfv 6540  ωcom 7851  reccrdg 8405  ℵcale 9927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-inf2 9632

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-om 7852  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-oi 9501  df-har 9548  df-card 9930  df-aleph 9931

This theorem is referenced by:  alephfplem4  10098  alephfp  10099