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Theorem unblem4 9297
Description: Lemma for unbnn 9298. 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 7855 . . . 4 ω ⊆ On
2 sstr 3985 . . . 4 ((𝐴 ⊆ ω ∧ ω ⊆ On) → 𝐴 ⊆ On)
31, 2mpan2 688 . . 3 (𝐴 ⊆ ω → 𝐴 ⊆ On)
43adantr 480 . 2 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → 𝐴 ⊆ On)
5 frfnom 8433 . . . 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 6639 . . . 4 (𝐹 Fn ω ↔ (rec((𝑥 ∈ V ↦ (𝐴 ∖ suc 𝑥)), 𝐴) ↾ ω) Fn ω)
85, 7mpbir 230 . . 3 𝐹 Fn ω
96unblem2 9295 . . . 4 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → (𝑧 ∈ ω → (𝐹𝑧) ∈ 𝐴))
109ralrimiv 3139 . . 3 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → ∀𝑧 ∈ ω (𝐹𝑧) ∈ 𝐴)
11 ffnfv 7113 . . . 4 (𝐹:ω⟶𝐴 ↔ (𝐹 Fn ω ∧ ∀𝑧 ∈ ω (𝐹𝑧) ∈ 𝐴))
1211biimpri 227 . . 3 ((𝐹 Fn ω ∧ ∀𝑧 ∈ ω (𝐹𝑧) ∈ 𝐴) → 𝐹:ω⟶𝐴)
138, 10, 12sylancr 586 . 2 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → 𝐹:ω⟶𝐴)
146unblem3 9296 . . 3 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → (𝑧 ∈ ω → (𝐹𝑧) ∈ (𝐹‘suc 𝑧)))
1514ralrimiv 3139 . 2 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → ∀𝑧 ∈ ω (𝐹𝑧) ∈ (𝐹‘suc 𝑧))
16 omsmo 8656 . 2 (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑧 ∈ ω (𝐹𝑧) ∈ (𝐹‘suc 𝑧)) → 𝐹:ω–1-1𝐴)
174, 13, 15, 16syl21anc 835 1 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → 𝐹:ω–1-1𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  wral 3055  wrex 3064  Vcvv 3468  cdif 3940  wss 3943   cint 4943  cmpt 5224  cres 5671  Oncon0 6357  suc csuc 6359   Fn wfn 6531  wf 6532  1-1wf1 6533  cfv 6536  ωcom 7851  reccrdg 8407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-om 7852  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408
This theorem is referenced by:  unbnn  9298
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