Theorem alephfplem3 10028

Description: Lemma for alephfp 10030. (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 6841	. . 3 ⊢ (𝑣 = ∅ → (𝐻‘𝑣) = (𝐻‘∅))
2	1	eleq1d 2822	. 2 ⊢ (𝑣 = ∅ → ((𝐻‘𝑣) ∈ ran ℵ ↔ (𝐻‘∅) ∈ ran ℵ))
3		fveq2 6841	. . 3 ⊢ (𝑣 = 𝑤 → (𝐻‘𝑣) = (𝐻‘𝑤))
4	3	eleq1d 2822	. 2 ⊢ (𝑣 = 𝑤 → ((𝐻‘𝑣) ∈ ran ℵ ↔ (𝐻‘𝑤) ∈ ran ℵ))
5		fveq2 6841	. . 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 10026	. 2 ⊢ (𝐻‘∅) ∈ ran ℵ
9		alephfnon 9987	. . . 4 ⊢ ℵ Fn On
10		alephsson 10022	. . . . 5 ⊢ ran ℵ ⊆ On
11	10	sseli 3918	. . . 4 ⊢ ((𝐻‘𝑤) ∈ ran ℵ → (𝐻‘𝑤) ∈ On)
12		fnfvelrn 7033	. . . 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 10027	. . . 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 7850	1 ⊢ (𝑣 ∈ ω → (𝐻‘𝑣) ∈ ran ℵ)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∅c0 4274  ran crn 5632   ↾ cres 5633  Oncon0 6324  suc csuc 6326   Fn wfn 6494  ‘cfv 6499  ωcom 7817  reccrdg 8348  ℵcale 9860

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 5213  ax-sep 5232  ax-nul 5242  ax-pow 5308  ax-pr 5376  ax-un 7689  ax-inf2 9562

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 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6266  df-ord 6327  df-on 6328  df-lim 6329  df-suc 6330  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7324  df-ov 7370  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-oi 9425  df-har 9472  df-card 9863  df-aleph 9864

This theorem is referenced by:  alephfplem4  10029  alephfp  10030