Theorem alephfplem4 10187

Description: Lemma for alephfp 10188. (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 8444	. . . . 5 ⊢ (rec(ℵ, ω) ↾ ω) Fn ω
2		alephfplem.1	. . . . . 6 ⊢ 𝐻 = (rec(ℵ, ω) ↾ ω)
3	2	fneq1i 6637	. . . . 5 ⊢ (𝐻 Fn ω ↔ (rec(ℵ, ω) ↾ ω) Fn ω)
4	1, 3	mpbir 234	. . . 4 ⊢ 𝐻 Fn ω
5	2	alephfplem3 10186	. . . . 5 ⊢ (𝑧 ∈ ω → (𝐻‘𝑧) ∈ ran ℵ)
6	5	rgen 3079	. . . 4 ⊢ ∀𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ
7		ffnfv 7120	. . . 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 4125	. . 3 ⊢ ran ℵ ⊆ (ω ∪ ran ℵ)
10		fss 6727	. . 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 7901	. . 3 ⊢ ∅ ∈ ω
13	2	alephfplem1 10184	. . 3 ⊢ (𝐻‘∅) ∈ ran ℵ
14		fveq2 6886	. . . . 5 ⊢ (𝑧 = ∅ → (𝐻‘𝑧) = (𝐻‘∅))
15	14	eleq1d 2846	. . . 4 ⊢ (𝑧 = ∅ → ((𝐻‘𝑧) ∈ ran ℵ ↔ (𝐻‘∅) ∈ ran ℵ))
16	15	rspcev 3577	. . 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 9645	. . 3 ⊢ ω ∈ V
19		cardinfima 10177	. . 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 3077  ∃wrex 3087  Vcvv 3451   ∪ cun 3897   ⊆ wss 3914  ∅c0 4279  ∪ cuni 4867  ran crn 5652   ↾ cres 5653   “ cima 5654   Fn wfn 6534  ⟶wf 6535  ‘cfv 6539  ωcom 7878  reccrdg 8418  ℵcale 10018

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 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7752  ax-inf2 9643

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3900  df-un 3902  df-in 3904  df-ss 3915  df-pss 4035  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6305  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6495  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-isom 6548  df-riota 7378  df-ov 7424  df-om 7879  df-2nd 8003  df-frecs 8300  df-wrecs 8331  df-recs 8380  df-rdg 8419  df-1o 8477  df-er 8718  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-oi 9505  df-har 9552  df-card 10021  df-aleph 10022

This theorem is used by:  alephfp  10188  alephfp2  10189