Theorem alephfplem3 10020

Description: Lemma for alephfp 10022. (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 6835	. . 3 ⊢ (𝑣 = ∅ → (𝐻‘𝑣) = (𝐻‘∅))
2	1	eleq1d 2822	. 2 ⊢ (𝑣 = ∅ → ((𝐻‘𝑣) ∈ ran ℵ ↔ (𝐻‘∅) ∈ ran ℵ))
3		fveq2 6835	. . 3 ⊢ (𝑣 = 𝑤 → (𝐻‘𝑣) = (𝐻‘𝑤))
4	3	eleq1d 2822	. 2 ⊢ (𝑣 = 𝑤 → ((𝐻‘𝑣) ∈ ran ℵ ↔ (𝐻‘𝑤) ∈ ran ℵ))
5		fveq2 6835	. . 3 ⊢ (𝑣 = suc 𝑤 → (𝐻‘𝑣) = (𝐻‘suc 𝑤))
6	5	eleq1d 2822	. 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 10018	. 2 ⊢ (𝐻‘∅) ∈ ran ℵ
9		alephfnon 9979	. . . 4 ⊢ ℵ Fn On
10		alephsson 10014	. . . . 5 ⊢ ran ℵ ⊆ On
11	10	sseli 3930	. . . 4 ⊢ ((𝐻‘𝑤) ∈ ran ℵ → (𝐻‘𝑤) ∈ On)
12		fnfvelrn 7027	. . . 4 ⊢ ((ℵ Fn On ∧ (𝐻‘𝑤) ∈ On) → (ℵ‘(𝐻‘𝑤)) ∈ ran ℵ)
13	9, 11, 12	sylancr 588	. . 3 ⊢ ((𝐻‘𝑤) ∈ ran ℵ → (ℵ‘(𝐻‘𝑤)) ∈ ran ℵ)
14	7	alephfplem2 10019	. . . 4 ⊢ (𝑤 ∈ ω → (𝐻‘suc 𝑤) = (ℵ‘(𝐻‘𝑤)))
15	14	eleq1d 2822	. . 3 ⊢ (𝑤 ∈ ω → ((𝐻‘suc 𝑤) ∈ ran ℵ ↔ (ℵ‘(𝐻‘𝑤)) ∈ ran ℵ))
16	13, 15	imbitrrid 246	. 2 ⊢ (𝑤 ∈ ω → ((𝐻‘𝑤) ∈ ran ℵ → (𝐻‘suc 𝑤) ∈ ran ℵ))
17	2, 4, 6, 8, 16	finds1 7843	1 ⊢ (𝑣 ∈ ω → (𝐻‘𝑣) ∈ ran ℵ)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∅c0 4286  ran crn 5626   ↾ cres 5627  Oncon0 6318  suc csuc 6320   Fn wfn 6488  ‘cfv 6493  ωcom 7810  reccrdg 8342  ℵcale 9852

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-inf2 9554

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7317  df-ov 7363  df-om 7811  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-oi 9419  df-har 9466  df-card 9855  df-aleph 9856

This theorem is referenced by:  alephfplem4  10021  alephfp  10022