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Theorem unblem4 9328
Description: Lemma for unbnn 9329. 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 7890 . . . 4 ω ⊆ On
2 sstr 4003 . . . 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 8473 . . . 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 9326 . . . 4 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → (𝑧 ∈ ω → (𝐹𝑧) ∈ 𝐴))
109ralrimiv 3142 . . 3 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → ∀𝑧 ∈ ω (𝐹𝑧) ∈ 𝐴)
11 ffnfv 7138 . . . 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 9327 . . 3 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → (𝑧 ∈ ω → (𝐹𝑧) ∈ (𝐹‘suc 𝑧)))
1514ralrimiv 3142 . 2 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → ∀𝑧 ∈ ω (𝐹𝑧) ∈ (𝐹‘suc 𝑧))
16 omsmo 8694 . 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 1536  wcel 2105  wral 3058  wrex 3067  Vcvv 3477  cdif 3959  wss 3962   cint 4950  cmpt 5230  cres 5690  Oncon0 6385  suc csuc 6387   Fn wfn 6557  wf 6558  1-1wf1 6559  cfv 6562  ωcom 7886  reccrdg 8447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448
This theorem is referenced by:  unbnn  9329
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