Theorem alephfplem4 10113

Description: Lemma for alephfp 10114. (Contributed by NM, 5-Nov-2004.)

Hypothesis

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

Assertion

Ref	Expression
alephfplem4	⊢ ∪ (𝐻 “ ω) ∈ ran ℵ

Proof of Theorem alephfplem4

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frfnom 8427	. . . . 5 ⊢ (rec(ℵ, ω) ↾ ω) Fn ω
2		alephfplem.1	. . . . . 6 ⊢ 𝐻 = (rec(ℵ, ω) ↾ ω)
3	2	fneq1i 6633	. . . . 5 ⊢ (𝐻 Fn ω ↔ (rec(ℵ, ω) ↾ ω) Fn ω)
4	1, 3	mpbir 234	. . . 4 ⊢ 𝐻 Fn ω
5	2	alephfplem3 10112	. . . . 5 ⊢ (𝑧 ∈ ω → (𝐻‘𝑧) ∈ ran ℵ)
6	5	rgen 3080	. . . 4 ⊢ ∀𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ
7		ffnfv 7115	. . . 4 ⊢ (𝐻:ω⟶ran ℵ ↔ (𝐻 Fn ω ∧ ∀𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ))
8	4, 6, 7	mpbir2an 724	. . 3 ⊢ 𝐻:ω⟶ran ℵ
9		ssun2 4128	. . 3 ⊢ ran ℵ ⊆ (ω ∪ ran ℵ)
10		fss 6723	. . 3 ⊢ ((𝐻:ω⟶ran ℵ ∧ ran ℵ ⊆ (ω ∪ ran ℵ)) → 𝐻:ω⟶(ω ∪ ran ℵ))
11	8, 9, 10	mp2an 705	. 2 ⊢ 𝐻:ω⟶(ω ∪ ran ℵ)
12		peano1 7888	. . 3 ⊢ ∅ ∈ ω
13	2	alephfplem1 10110	. . 3 ⊢ (𝐻‘∅) ∈ ran ℵ
14		fveq2 6882	. . . . 5 ⊢ (𝑧 = ∅ → (𝐻‘𝑧) = (𝐻‘∅))
15	14	eleq1d 2847	. . . 4 ⊢ (𝑧 = ∅ → ((𝐻‘𝑧) ∈ ran ℵ ↔ (𝐻‘∅) ∈ ran ℵ))
16	15	rspcev 3579	. . 3 ⊢ ((∅ ∈ ω ∧ (𝐻‘∅) ∈ ran ℵ) → ∃𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ)
17	12, 13, 16	mp2an 705	. 2 ⊢ ∃𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ
18		omex 9625	. . 3 ⊢ ω ∈ V
19		cardinfima 10103	. . 3 ⊢ (ω ∈ V → ((𝐻:ω⟶(ω ∪ ran ℵ) ∧ ∃𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ) → ∪ (𝐻 “ ω) ∈ ran ℵ))
20	18, 19	ax-mp 5	. 2 ⊢ ((𝐻:ω⟶(ω ∪ ran ℵ) ∧ ∃𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ) → ∪ (𝐻 “ ω) ∈ ran ℵ)
21	11, 17, 20	mp2an 705	1 ⊢ ∪ (𝐻 “ ω) ∈ ran ℵ

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  ∃wrex 3088  Vcvv 3453   ∪ cun 3900   ⊆ wss 3902  ∅c0 4282  ∪ cuni 4870  ran crn 5660   ↾ cres 5661   “ cima 5662   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  ωcom 7865  reccrdg 8401  ℵcale 9944

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-inf2 9623

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7373  df-ov 7419  df-om 7866  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-er 8699  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-oi 9485  df-har 9532  df-card 9947  df-aleph 9948

This theorem is used by:  alephfp  10114  alephfp2  10115