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Theorem unblem4 8256
Description: Lemma for unbnn 8257. The function 𝐹 maps the set of natural numbers one-to-one to the set of unbounded natural numbers 𝐴. (Contributed by NM, 3-Dec-2003.)
Hypothesis
Ref Expression
unblem.2 𝐹 = (rec((𝑥 ∈ V ↦ (𝐴 ∖ suc 𝑥)), 𝐴) ↾ ω)
Assertion
Ref Expression
unblem4 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → 𝐹:ω–1-1𝐴)
Distinct variable groups:   𝑤,𝑣,𝑥,𝐴   𝑣,𝐹,𝑤
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem unblem4
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 omsson 7111 . . . 4 ω ⊆ On
2 sstr 3644 . . . 4 ((𝐴 ⊆ ω ∧ ω ⊆ On) → 𝐴 ⊆ On)
31, 2mpan2 707 . . 3 (𝐴 ⊆ ω → 𝐴 ⊆ On)
43adantr 480 . 2 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → 𝐴 ⊆ On)
5 frfnom 7575 . . . 4 (rec((𝑥 ∈ V ↦ (𝐴 ∖ suc 𝑥)), 𝐴) ↾ ω) Fn ω
6 unblem.2 . . . . 5 𝐹 = (rec((𝑥 ∈ V ↦ (𝐴 ∖ suc 𝑥)), 𝐴) ↾ ω)
76fneq1i 6023 . . . 4 (𝐹 Fn ω ↔ (rec((𝑥 ∈ V ↦ (𝐴 ∖ suc 𝑥)), 𝐴) ↾ ω) Fn ω)
85, 7mpbir 221 . . 3 𝐹 Fn ω
96unblem2 8254 . . . 4 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → (𝑧 ∈ ω → (𝐹𝑧) ∈ 𝐴))
109ralrimiv 2994 . . 3 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → ∀𝑧 ∈ ω (𝐹𝑧) ∈ 𝐴)
11 ffnfv 6428 . . . 4 (𝐹:ω⟶𝐴 ↔ (𝐹 Fn ω ∧ ∀𝑧 ∈ ω (𝐹𝑧) ∈ 𝐴))
1211biimpri 218 . . 3 ((𝐹 Fn ω ∧ ∀𝑧 ∈ ω (𝐹𝑧) ∈ 𝐴) → 𝐹:ω⟶𝐴)
138, 10, 12sylancr 696 . 2 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → 𝐹:ω⟶𝐴)
146unblem3 8255 . . 3 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → (𝑧 ∈ ω → (𝐹𝑧) ∈ (𝐹‘suc 𝑧)))
1514ralrimiv 2994 . 2 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → ∀𝑧 ∈ ω (𝐹𝑧) ∈ (𝐹‘suc 𝑧))
16 omsmo 7779 . 2 (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑧 ∈ ω (𝐹𝑧) ∈ (𝐹‘suc 𝑧)) → 𝐹:ω–1-1𝐴)
174, 13, 15, 16syl21anc 1365 1 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → 𝐹:ω–1-1𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  Vcvv 3231  cdif 3604  wss 3607   cint 4507  cmpt 4762  cres 5145  Oncon0 5761  suc csuc 5763   Fn wfn 5921  wf 5922  1-1wf1 5923  cfv 5926  ωcom 7107  reccrdg 7550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551
This theorem is referenced by:  unbnn  8257
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