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Theorem alephfplem4 10141
Description: Lemma for alephfp 10142. (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 8455 . . . . 5 (rec(ℵ, ω) ↾ ω) Fn ω
2 alephfplem.1 . . . . . 6 𝐻 = (rec(ℵ, ω) ↾ ω)
32fneq1i 6647 . . . . 5 (𝐻 Fn ω ↔ (rec(ℵ, ω) ↾ ω) Fn ω)
41, 3mpbir 230 . . . 4 𝐻 Fn ω
52alephfplem3 10140 . . . . 5 (𝑧 ∈ ω → (𝐻𝑧) ∈ ran ℵ)
65rgen 3053 . . . 4 𝑧 ∈ ω (𝐻𝑧) ∈ ran ℵ
7 ffnfv 7123 . . . 4 (𝐻:ω⟶ran ℵ ↔ (𝐻 Fn ω ∧ ∀𝑧 ∈ ω (𝐻𝑧) ∈ ran ℵ))
84, 6, 7mpbir2an 709 . . 3 𝐻:ω⟶ran ℵ
9 ssun2 4172 . . 3 ran ℵ ⊆ (ω ∪ ran ℵ)
10 fss 6734 . . 3 ((𝐻:ω⟶ran ℵ ∧ ran ℵ ⊆ (ω ∪ ran ℵ)) → 𝐻:ω⟶(ω ∪ ran ℵ))
118, 9, 10mp2an 690 . 2 𝐻:ω⟶(ω ∪ ran ℵ)
12 peano1 7890 . . 3 ∅ ∈ ω
132alephfplem1 10138 . . 3 (𝐻‘∅) ∈ ran ℵ
14 fveq2 6891 . . . . 5 (𝑧 = ∅ → (𝐻𝑧) = (𝐻‘∅))
1514eleq1d 2811 . . . 4 (𝑧 = ∅ → ((𝐻𝑧) ∈ ran ℵ ↔ (𝐻‘∅) ∈ ran ℵ))
1615rspcev 3608 . . 3 ((∅ ∈ ω ∧ (𝐻‘∅) ∈ ran ℵ) → ∃𝑧 ∈ ω (𝐻𝑧) ∈ ran ℵ)
1712, 13, 16mp2an 690 . 2 𝑧 ∈ ω (𝐻𝑧) ∈ ran ℵ
18 omex 9677 . . 3 ω ∈ V
19 cardinfima 10131 . . 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 690 1 (𝐻 “ ω) ∈ ran ℵ
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  wral 3051  wrex 3060  Vcvv 3463  cun 3945  wss 3947  c0 4323   cuni 4906  ran crn 5674  cres 5675  cima 5676   Fn wfn 6539  wf 6540  cfv 6544  ωcom 7866  reccrdg 8429  cale 9970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7736  ax-inf2 9675
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3365  df-reu 3366  df-rab 3421  df-v 3465  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4324  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-int 4948  df-iun 4996  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-se 5629  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6303  df-ord 6369  df-on 6370  df-lim 6371  df-suc 6372  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7370  df-ov 7417  df-om 7867  df-2nd 7994  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-1o 8486  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-oi 9544  df-har 9591  df-card 9973  df-aleph 9974
This theorem is referenced by:  alephfp  10142  alephfp2  10143
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