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Theorem unblem4 9331
Description: Lemma for unbnn 9332. 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 7891 . . . 4 ω ⊆ On
2 sstr 3992 . . . 4 ((𝐴 ⊆ ω ∧ ω ⊆ On) → 𝐴 ⊆ On)
31, 2mpan2 691 . . 3 (𝐴 ⊆ ω → 𝐴 ⊆ On)
43adantr 480 . 2 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → 𝐴 ⊆ On)
5 frfnom 8475 . . . 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 6665 . . . 4 (𝐹 Fn ω ↔ (rec((𝑥 ∈ V ↦ (𝐴 ∖ suc 𝑥)), 𝐴) ↾ ω) Fn ω)
85, 7mpbir 231 . . 3 𝐹 Fn ω
96unblem2 9329 . . . 4 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → (𝑧 ∈ ω → (𝐹𝑧) ∈ 𝐴))
109ralrimiv 3145 . . 3 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → ∀𝑧 ∈ ω (𝐹𝑧) ∈ 𝐴)
11 ffnfv 7139 . . . 4 (𝐹:ω⟶𝐴 ↔ (𝐹 Fn ω ∧ ∀𝑧 ∈ ω (𝐹𝑧) ∈ 𝐴))
1211biimpri 228 . . 3 ((𝐹 Fn ω ∧ ∀𝑧 ∈ ω (𝐹𝑧) ∈ 𝐴) → 𝐹:ω⟶𝐴)
138, 10, 12sylancr 587 . 2 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → 𝐹:ω⟶𝐴)
146unblem3 9330 . . 3 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → (𝑧 ∈ ω → (𝐹𝑧) ∈ (𝐹‘suc 𝑧)))
1514ralrimiv 3145 . 2 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → ∀𝑧 ∈ ω (𝐹𝑧) ∈ (𝐹‘suc 𝑧))
16 omsmo 8696 . 2 (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑧 ∈ ω (𝐹𝑧) ∈ (𝐹‘suc 𝑧)) → 𝐹:ω–1-1𝐴)
174, 13, 15, 16syl21anc 838 1 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → 𝐹:ω–1-1𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061  wrex 3070  Vcvv 3480  cdif 3948  wss 3951   cint 4946  cmpt 5225  cres 5687  Oncon0 6384  suc csuc 6386   Fn wfn 6556  wf 6557  1-1wf1 6558  cfv 6561  ωcom 7887  reccrdg 8449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450
This theorem is referenced by:  unbnn  9332
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