Theorem alephfplem4 9863

Description: Lemma for alephfp 9864. (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 8266	. . . . 5 ⊢ (rec(ℵ, ω) ↾ ω) Fn ω
2		alephfplem.1	. . . . . 6 ⊢ 𝐻 = (rec(ℵ, ω) ↾ ω)
3	2	fneq1i 6530	. . . . 5 ⊢ (𝐻 Fn ω ↔ (rec(ℵ, ω) ↾ ω) Fn ω)
4	1, 3	mpbir 230	. . . 4 ⊢ 𝐻 Fn ω
5	2	alephfplem3 9862	. . . . 5 ⊢ (𝑧 ∈ ω → (𝐻‘𝑧) ∈ ran ℵ)
6	5	rgen 3074	. . . 4 ⊢ ∀𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ
7		ffnfv 6992	. . . 4 ⊢ (𝐻:ω⟶ran ℵ ↔ (𝐻 Fn ω ∧ ∀𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ))
8	4, 6, 7	mpbir2an 708	. . 3 ⊢ 𝐻:ω⟶ran ℵ
9		ssun2 4107	. . 3 ⊢ ran ℵ ⊆ (ω ∪ ran ℵ)
10		fss 6617	. . 3 ⊢ ((𝐻:ω⟶ran ℵ ∧ ran ℵ ⊆ (ω ∪ ran ℵ)) → 𝐻:ω⟶(ω ∪ ran ℵ))
11	8, 9, 10	mp2an 689	. 2 ⊢ 𝐻:ω⟶(ω ∪ ran ℵ)
12		peano1 7735	. . 3 ⊢ ∅ ∈ ω
13	2	alephfplem1 9860	. . 3 ⊢ (𝐻‘∅) ∈ ran ℵ
14		fveq2 6774	. . . . 5 ⊢ (𝑧 = ∅ → (𝐻‘𝑧) = (𝐻‘∅))
15	14	eleq1d 2823	. . . 4 ⊢ (𝑧 = ∅ → ((𝐻‘𝑧) ∈ ran ℵ ↔ (𝐻‘∅) ∈ ran ℵ))
16	15	rspcev 3561	. . 3 ⊢ ((∅ ∈ ω ∧ (𝐻‘∅) ∈ ran ℵ) → ∃𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ)
17	12, 13, 16	mp2an 689	. 2 ⊢ ∃𝑧 ∈ ω (𝐻‘𝑧) ∈ ran ℵ
18		omex 9401	. . 3 ⊢ ω ∈ V
19		cardinfima 9853	. . 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 689	1 ⊢ ∪ (𝐻 “ ω) ∈ ran ℵ

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  Vcvv 3432   ∪ cun 3885   ⊆ wss 3887  ∅c0 4256  ∪ cuni 4839  ran crn 5590   ↾ cres 5591   “ cima 5592   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  ωcom 7712  reccrdg 8240  ℵcale 9694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-oi 9269  df-har 9316  df-card 9697  df-aleph 9698

This theorem is referenced by:  alephfp  9864  alephfp2  9865