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Theorem alephfplem4 10002
Description: Lemma for alephfp 10003. (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 8374 . . . . 5 (rec(ℵ, ω) ↾ ω) Fn ω
2 alephfplem.1 . . . . . 6 𝐻 = (rec(ℵ, ω) ↾ ω)
32fneq1i 6597 . . . . 5 (𝐻 Fn ω ↔ (rec(ℵ, ω) ↾ ω) Fn ω)
41, 3mpbir 230 . . . 4 𝐻 Fn ω
52alephfplem3 10001 . . . . 5 (𝑧 ∈ ω → (𝐻𝑧) ∈ ran ℵ)
65rgen 3065 . . . 4 𝑧 ∈ ω (𝐻𝑧) ∈ ran ℵ
7 ffnfv 7063 . . . 4 (𝐻:ω⟶ran ℵ ↔ (𝐻 Fn ω ∧ ∀𝑧 ∈ ω (𝐻𝑧) ∈ ran ℵ))
84, 6, 7mpbir2an 710 . . 3 𝐻:ω⟶ran ℵ
9 ssun2 4132 . . 3 ran ℵ ⊆ (ω ∪ ran ℵ)
10 fss 6683 . . 3 ((𝐻:ω⟶ran ℵ ∧ ran ℵ ⊆ (ω ∪ ran ℵ)) → 𝐻:ω⟶(ω ∪ ran ℵ))
118, 9, 10mp2an 691 . 2 𝐻:ω⟶(ω ∪ ran ℵ)
12 peano1 7818 . . 3 ∅ ∈ ω
132alephfplem1 9999 . . 3 (𝐻‘∅) ∈ ran ℵ
14 fveq2 6840 . . . . 5 (𝑧 = ∅ → (𝐻𝑧) = (𝐻‘∅))
1514eleq1d 2823 . . . 4 (𝑧 = ∅ → ((𝐻𝑧) ∈ ran ℵ ↔ (𝐻‘∅) ∈ ran ℵ))
1615rspcev 3580 . . 3 ((∅ ∈ ω ∧ (𝐻‘∅) ∈ ran ℵ) → ∃𝑧 ∈ ω (𝐻𝑧) ∈ ran ℵ)
1712, 13, 16mp2an 691 . 2 𝑧 ∈ ω (𝐻𝑧) ∈ ran ℵ
18 omex 9538 . . 3 ω ∈ V
19 cardinfima 9992 . . 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 691 1 (𝐻 “ ω) ∈ ran ℵ
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3063  wrex 3072  Vcvv 3444  cun 3907  wss 3909  c0 4281   cuni 4864  ran crn 5633  cres 5634  cima 5635   Fn wfn 6489  wf 6490  cfv 6494  ωcom 7795  reccrdg 8348  cale 9831
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-rep 5241  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7665  ax-inf2 9536
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-pss 3928  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-int 4907  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-tr 5222  df-id 5530  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5587  df-se 5588  df-we 5589  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6252  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-isom 6503  df-riota 7308  df-ov 7355  df-om 7796  df-2nd 7915  df-frecs 8205  df-wrecs 8236  df-recs 8310  df-rdg 8349  df-1o 8405  df-er 8607  df-en 8843  df-dom 8844  df-sdom 8845  df-fin 8846  df-oi 9405  df-har 9452  df-card 9834  df-aleph 9835
This theorem is referenced by:  alephfp  10003  alephfp2  10004
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