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Theorem unblem4 8077
Description: Lemma for unbnn 8078. 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 6938 . . . 4 ω ⊆ On
2 sstr 3575 . . . 4 ((𝐴 ⊆ ω ∧ ω ⊆ On) → 𝐴 ⊆ On)
31, 2mpan2 702 . . 3 (𝐴 ⊆ ω → 𝐴 ⊆ On)
43adantr 479 . 2 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → 𝐴 ⊆ On)
5 frfnom 7394 . . . 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 5885 . . . 4 (𝐹 Fn ω ↔ (rec((𝑥 ∈ V ↦ (𝐴 ∖ suc 𝑥)), 𝐴) ↾ ω) Fn ω)
85, 7mpbir 219 . . 3 𝐹 Fn ω
96unblem2 8075 . . . 4 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → (𝑧 ∈ ω → (𝐹𝑧) ∈ 𝐴))
109ralrimiv 2947 . . 3 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → ∀𝑧 ∈ ω (𝐹𝑧) ∈ 𝐴)
11 ffnfv 6280 . . . 4 (𝐹:ω⟶𝐴 ↔ (𝐹 Fn ω ∧ ∀𝑧 ∈ ω (𝐹𝑧) ∈ 𝐴))
1211biimpri 216 . . 3 ((𝐹 Fn ω ∧ ∀𝑧 ∈ ω (𝐹𝑧) ∈ 𝐴) → 𝐹:ω⟶𝐴)
138, 10, 12sylancr 693 . 2 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → 𝐹:ω⟶𝐴)
146unblem3 8076 . . 3 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → (𝑧 ∈ ω → (𝐹𝑧) ∈ (𝐹‘suc 𝑧)))
1514ralrimiv 2947 . 2 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → ∀𝑧 ∈ ω (𝐹𝑧) ∈ (𝐹‘suc 𝑧))
16 omsmo 7598 . 2 (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑧 ∈ ω (𝐹𝑧) ∈ (𝐹‘suc 𝑧)) → 𝐹:ω–1-1𝐴)
174, 13, 15, 16syl21anc 1316 1 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → 𝐹:ω–1-1𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  wral 2895  wrex 2896  Vcvv 3172  cdif 3536  wss 3539   cint 4404  cmpt 4637  cres 5030  Oncon0 5626  suc csuc 5628   Fn wfn 5785  wf 5786  1-1wf1 5787  cfv 5790  ωcom 6934  reccrdg 7369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-om 6935  df-wrecs 7271  df-recs 7332  df-rdg 7370
This theorem is referenced by:  unbnn  8078
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