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Theorem alephfplem4 9216
Description: Lemma for alephfp 9217. (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 7769 . . . . 5 (rec(ℵ, ω) ↾ ω) Fn ω
2 alephfplem.1 . . . . . 6 𝐻 = (rec(ℵ, ω) ↾ ω)
32fneq1i 6196 . . . . 5 (𝐻 Fn ω ↔ (rec(ℵ, ω) ↾ ω) Fn ω)
41, 3mpbir 223 . . . 4 𝐻 Fn ω
52alephfplem3 9215 . . . . 5 (𝑧 ∈ ω → (𝐻𝑧) ∈ ran ℵ)
65rgen 3103 . . . 4 𝑧 ∈ ω (𝐻𝑧) ∈ ran ℵ
7 ffnfv 6614 . . . 4 (𝐻:ω⟶ran ℵ ↔ (𝐻 Fn ω ∧ ∀𝑧 ∈ ω (𝐻𝑧) ∈ ran ℵ))
84, 6, 7mpbir2an 703 . . 3 𝐻:ω⟶ran ℵ
9 ssun2 3975 . . 3 ran ℵ ⊆ (ω ∪ ran ℵ)
10 fss 6269 . . 3 ((𝐻:ω⟶ran ℵ ∧ ran ℵ ⊆ (ω ∪ ran ℵ)) → 𝐻:ω⟶(ω ∪ ran ℵ))
118, 9, 10mp2an 684 . 2 𝐻:ω⟶(ω ∪ ran ℵ)
12 peano1 7319 . . 3 ∅ ∈ ω
132alephfplem1 9213 . . 3 (𝐻‘∅) ∈ ran ℵ
14 fveq2 6411 . . . . 5 (𝑧 = ∅ → (𝐻𝑧) = (𝐻‘∅))
1514eleq1d 2863 . . . 4 (𝑧 = ∅ → ((𝐻𝑧) ∈ ran ℵ ↔ (𝐻‘∅) ∈ ran ℵ))
1615rspcev 3497 . . 3 ((∅ ∈ ω ∧ (𝐻‘∅) ∈ ran ℵ) → ∃𝑧 ∈ ω (𝐻𝑧) ∈ ran ℵ)
1712, 13, 16mp2an 684 . 2 𝑧 ∈ ω (𝐻𝑧) ∈ ran ℵ
18 omex 8790 . . 3 ω ∈ V
19 cardinfima 9206 . . 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 684 1 (𝐻 “ ω) ∈ ran ℵ
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  wral 3089  wrex 3090  Vcvv 3385  cun 3767  wss 3769  c0 4115   cuni 4628  ran crn 5313  cres 5314  cima 5315   Fn wfn 6096  wf 6097  cfv 6101  ωcom 7299  reccrdg 7744  cale 9048
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-inf2 8788
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-se 5272  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-isom 6110  df-riota 6839  df-om 7300  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-oi 8657  df-har 8705  df-card 9051  df-aleph 9052
This theorem is referenced by:  alephfp  9217  alephfp2  9218
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