Theorem alephfplem3 4890

Description: Lemma for alephfp 4892.

Hypothesis

Ref	Expression
alephfplem.1	⊢ H = (rec({⟨x, y⟩∣y = (ℵ ‘x)}, (ℵ ‘∅)) ↾ ω)

Assertion

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

Distinct variable groups:   x,y,v   v,H

Proof of Theorem alephfplem3

Step	Hyp	Ref	Expression
1		equid 1124	. 2 ⊢ y = y
2		fveq2 3726	. . . 4 ⊢ (v = ∅ → (H ‘v) = (H ‘∅))
3	2	eleq1d 1537	. . 3 ⊢ (v = ∅ → ((H ‘v) ∈ ran ℵ ↔ (H ‘∅) ∈ ran ℵ))
4		fveq2 3726	. . . 4 ⊢ (v = w → (H ‘v) = (H ‘w))
5	4	eleq1d 1537	. . 3 ⊢ (v = w → ((H ‘v) ∈ ran ℵ ↔ (H ‘w) ∈ ran ℵ))
6		fveq2 3726	. . . 4 ⊢ (v = suc w → (H ‘v) = (H ‘suc w))
7	6	eleq1d 1537	. . 3 ⊢ (v = suc w → ((H ‘v) ∈ ran ℵ ↔ (H ‘suc w) ∈ ran ℵ))
8		alephfplem.1	. . . . 5 ⊢ H = (rec({⟨x, y⟩∣y = (ℵ ‘x)}, (ℵ ‘∅)) ↾ ω)
9	8	alephfplem1 4888	. . . 4 ⊢ (H ‘∅) ∈ ran ℵ
10	9	a1i 8	. . 3 ⊢ (y = y → (H ‘∅) ∈ ran ℵ)
11	8	alephfplem2 4889	. . . . . 6 ⊢ (w ∈ ω → (H ‘suc w) = (ℵ ‘(H ‘w)))
12	11	eleq1d 1537	. . . . 5 ⊢ (w ∈ ω → ((H ‘suc w) ∈ ran ℵ ↔ (ℵ ‘(H ‘w)) ∈ ran ℵ))
13		alephsson 4886	. . . . . . 7 ⊢ ran ℵ ⊆ On
14	13	sseli 2061	. . . . . 6 ⊢ ((H ‘w) ∈ ran ℵ → (H ‘w) ∈ On)
15		alephfnon 4854	. . . . . . 7 ⊢ ℵ Fn On
16		fnfvelrn 3815	. . . . . . 7 ⊢ ((ℵ Fn On ⋀ (H ‘w) ∈ On) → (ℵ ‘(H ‘w)) ∈ ran ℵ)
17	15, 16	mpan 694	. . . . . 6 ⊢ ((H ‘w) ∈ On → (ℵ ‘(H ‘w)) ∈ ran ℵ)
18	14, 17	syl 10	. . . . 5 ⊢ ((H ‘w) ∈ ran ℵ → (ℵ ‘(H ‘w)) ∈ ran ℵ)
19	12, 18	syl5bir 210	. . . 4 ⊢ (w ∈ ω → ((H ‘w) ∈ ran ℵ → (H ‘suc w) ∈ ran ℵ))
20	19	a1d 12	. . 3 ⊢ (w ∈ ω → (y = y → ((H ‘w) ∈ ran ℵ → (H ‘suc w) ∈ ran ℵ)))
21	3, 5, 7, 10, 20	finds2 3158	. 2 ⊢ (v ∈ ω → (y = y → (H ‘v) ∈ ran ℵ))
22	1, 21	mpi 44	1 ⊢ (v ∈ ω → (H ‘v) ∈ ran ℵ)

Syntax hints:   → wi 3   = wceq 954   ∈ wcel 956  ∅c0 2276  {copab 2662  Oncon0 2947  suc csuc 2949  ωcom 3131  ran crn 3171   ↾ cres 3172   Fn wfn 3177   ‘cfv 3182  reccrdg 3933  ℵcale 4806

This theorem is referenced by:  alephfplem4 4891  alephfp 4892

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2865  ax-reg 4585  ax-inf2 4617  ax-ac 4736

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2500  df-int 2530  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2829  df-id 2832  df-po 2839  df-so 2849  df-fr 2916  df-we 2933  df-ord 2950  df-on 2951  df-lim 2952  df-suc 2953  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-rdg 3934  df-er 4262  df-en 4368  df-dom 4369  df-sdom 4370  df-card 4808  df-aleph 4809

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