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Theorem alephfplem4 9386
Description: Lemma for alephfp 9387. (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 7929 . . . . 5 (rec(ℵ, ω) ↾ ω) Fn ω
2 alephfplem.1 . . . . . 6 𝐻 = (rec(ℵ, ω) ↾ ω)
32fneq1i 6327 . . . . 5 (𝐻 Fn ω ↔ (rec(ℵ, ω) ↾ ω) Fn ω)
41, 3mpbir 232 . . . 4 𝐻 Fn ω
52alephfplem3 9385 . . . . 5 (𝑧 ∈ ω → (𝐻𝑧) ∈ ran ℵ)
65rgen 3117 . . . 4 𝑧 ∈ ω (𝐻𝑧) ∈ ran ℵ
7 ffnfv 6752 . . . 4 (𝐻:ω⟶ran ℵ ↔ (𝐻 Fn ω ∧ ∀𝑧 ∈ ω (𝐻𝑧) ∈ ran ℵ))
84, 6, 7mpbir2an 707 . . 3 𝐻:ω⟶ran ℵ
9 ssun2 4076 . . 3 ran ℵ ⊆ (ω ∪ ran ℵ)
10 fss 6402 . . 3 ((𝐻:ω⟶ran ℵ ∧ ran ℵ ⊆ (ω ∪ ran ℵ)) → 𝐻:ω⟶(ω ∪ ran ℵ))
118, 9, 10mp2an 688 . 2 𝐻:ω⟶(ω ∪ ran ℵ)
12 peano1 7464 . . 3 ∅ ∈ ω
132alephfplem1 9383 . . 3 (𝐻‘∅) ∈ ran ℵ
14 fveq2 6545 . . . . 5 (𝑧 = ∅ → (𝐻𝑧) = (𝐻‘∅))
1514eleq1d 2869 . . . 4 (𝑧 = ∅ → ((𝐻𝑧) ∈ ran ℵ ↔ (𝐻‘∅) ∈ ran ℵ))
1615rspcev 3561 . . 3 ((∅ ∈ ω ∧ (𝐻‘∅) ∈ ran ℵ) → ∃𝑧 ∈ ω (𝐻𝑧) ∈ ran ℵ)
1712, 13, 16mp2an 688 . 2 𝑧 ∈ ω (𝐻𝑧) ∈ ran ℵ
18 omex 8959 . . 3 ω ∈ V
19 cardinfima 9376 . . 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 688 1 (𝐻 “ ω) ∈ ran ℵ
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1525  wcel 2083  wral 3107  wrex 3108  Vcvv 3440  cun 3863  wss 3865  c0 4217   cuni 4751  ran crn 5451  cres 5452  cima 5453   Fn wfn 6227  wf 6228  cfv 6232  ωcom 7443  reccrdg 7904  cale 9218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-inf2 8957
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-int 4789  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-se 5410  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-isom 6241  df-riota 6984  df-om 7444  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-er 8146  df-en 8365  df-dom 8366  df-sdom 8367  df-fin 8368  df-oi 8827  df-har 8875  df-card 9221  df-aleph 9222
This theorem is referenced by:  alephfp  9387  alephfp2  9388
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