Theorem alephfplem4 9794

Description: Lemma for alephfp 9795. (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 8236	. . . . 5 ⊢ (rec(ℵ, ω) ↾ ω) Fn ω
2		alephfplem.1	. . . . . 6 ⊢ 𝐻 = (rec(ℵ, ω) ↾ ω)
3	2	fneq1i 6514	. . . . 5 ⊢ (𝐻 Fn ω ↔ (rec(ℵ, ω) ↾ ω) Fn ω)
4	1, 3	mpbir 230	. . . 4 ⊢ 𝐻 Fn ω
5	2	alephfplem3 9793	. . . . 5 ⊢ (𝑧 ∈ ω → (𝐻‘𝑧) ∈ ran ℵ)
6	5	rgen 3073	. . . 4 ⊢ ∀𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ
7		ffnfv 6974	. . . 4 ⊢ (𝐻:ω⟶ran ℵ ↔ (𝐻 Fn ω ∧ ∀𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ))
8	4, 6, 7	mpbir2an 707	. . 3 ⊢ 𝐻:ω⟶ran ℵ
9		ssun2 4103	. . 3 ⊢ ran ℵ ⊆ (ω ∪ ran ℵ)
10		fss 6601	. . 3 ⊢ ((𝐻:ω⟶ran ℵ ∧ ran ℵ ⊆ (ω ∪ ran ℵ)) → 𝐻:ω⟶(ω ∪ ran ℵ))
11	8, 9, 10	mp2an 688	. 2 ⊢ 𝐻:ω⟶(ω ∪ ran ℵ)
12		peano1 7710	. . 3 ⊢ ∅ ∈ ω
13	2	alephfplem1 9791	. . 3 ⊢ (𝐻‘∅) ∈ ran ℵ
14		fveq2 6756	. . . . 5 ⊢ (𝑧 = ∅ → (𝐻‘𝑧) = (𝐻‘∅))
15	14	eleq1d 2823	. . . 4 ⊢ (𝑧 = ∅ → ((𝐻‘𝑧) ∈ ran ℵ ↔ (𝐻‘∅) ∈ ran ℵ))
16	15	rspcev 3552	. . 3 ⊢ ((∅ ∈ ω ∧ (𝐻‘∅) ∈ ran ℵ) → ∃𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ)
17	12, 13, 16	mp2an 688	. 2 ⊢ ∃𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ
18		omex 9331	. . 3 ⊢ ω ∈ V
19		cardinfima 9784	. . 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 688	1 ⊢ ∪ (𝐻 “ ω) ∈ ran ℵ

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ∪ cun 3881   ⊆ wss 3883  ∅c0 4253  ∪ cuni 4836  ran crn 5581   ↾ cres 5582   “ cima 5583   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  ωcom 7687  reccrdg 8211  ℵcale 9625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-oi 9199  df-har 9246  df-card 9628  df-aleph 9629

This theorem is referenced by:  alephfp  9795  alephfp2  9796