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Theorem alephfplem4 8915
 Description: Lemma for alephfp 8916. (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.
StepHypRef Expression
1 frfnom 7515 . . . . 5 (rec(ℵ, ω) ↾ ω) Fn ω
2 alephfplem.1 . . . . . 6 𝐻 = (rec(ℵ, ω) ↾ ω)
32fneq1i 5973 . . . . 5 (𝐻 Fn ω ↔ (rec(ℵ, ω) ↾ ω) Fn ω)
41, 3mpbir 221 . . . 4 𝐻 Fn ω
52alephfplem3 8914 . . . . 5 (𝑧 ∈ ω → (𝐻𝑧) ∈ ran ℵ)
65rgen 2919 . . . 4 𝑧 ∈ ω (𝐻𝑧) ∈ ran ℵ
7 ffnfv 6374 . . . 4 (𝐻:ω⟶ran ℵ ↔ (𝐻 Fn ω ∧ ∀𝑧 ∈ ω (𝐻𝑧) ∈ ran ℵ))
84, 6, 7mpbir2an 954 . . 3 𝐻:ω⟶ran ℵ
9 ssun2 3769 . . 3 ran ℵ ⊆ (ω ∪ ran ℵ)
10 fss 6043 . . 3 ((𝐻:ω⟶ran ℵ ∧ ran ℵ ⊆ (ω ∪ ran ℵ)) → 𝐻:ω⟶(ω ∪ ran ℵ))
118, 9, 10mp2an 707 . 2 𝐻:ω⟶(ω ∪ ran ℵ)
12 peano1 7070 . . 3 ∅ ∈ ω
132alephfplem1 8912 . . 3 (𝐻‘∅) ∈ ran ℵ
14 fveq2 6178 . . . . 5 (𝑧 = ∅ → (𝐻𝑧) = (𝐻‘∅))
1514eleq1d 2684 . . . 4 (𝑧 = ∅ → ((𝐻𝑧) ∈ ran ℵ ↔ (𝐻‘∅) ∈ ran ℵ))
1615rspcev 3304 . . 3 ((∅ ∈ ω ∧ (𝐻‘∅) ∈ ran ℵ) → ∃𝑧 ∈ ω (𝐻𝑧) ∈ ran ℵ)
1712, 13, 16mp2an 707 . 2 𝑧 ∈ ω (𝐻𝑧) ∈ ran ℵ
18 omex 8525 . . 3 ω ∈ V
19 cardinfima 8905 . . 3 (ω ∈ V → ((𝐻:ω⟶(ω ∪ ran ℵ) ∧ ∃𝑧 ∈ ω (𝐻𝑧) ∈ ran ℵ) → (𝐻 “ ω) ∈ ran ℵ))
2018, 19ax-mp 5 . 2 ((𝐻:ω⟶(ω ∪ ran ℵ) ∧ ∃𝑧 ∈ ω (𝐻𝑧) ∈ ran ℵ) → (𝐻 “ ω) ∈ ran ℵ)
2111, 17, 20mp2an 707 1 (𝐻 “ ω) ∈ ran ℵ
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1481   ∈ wcel 1988  ∀wral 2909  ∃wrex 2910  Vcvv 3195   ∪ cun 3565   ⊆ wss 3567  ∅c0 3907  ∪ cuni 4427  ran crn 5105   ↾ cres 5106   “ cima 5107   Fn wfn 5871  ⟶wf 5872  ‘cfv 5876  ωcom 7050  reccrdg 7490  ℵcale 8747 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-om 7051  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-oi 8400  df-har 8448  df-card 8750  df-aleph 8751 This theorem is referenced by:  alephfp  8916  alephfp2  8917
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