Theorem unblem4 9262

Description: Lemma for unbnn 9263. 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 ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → 𝐹:ω–1-1→𝐴)

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.

Step	Hyp	Ref	Expression
1		omsson 7872	. . . 4 ⊢ ω ⊆ On
2		sstr 3946	. . . 4 ⊢ ((𝐴 ⊆ ω ∧ ω ⊆ On) → 𝐴 ⊆ On)
3	1, 2	mpan2 704	. . 3 ⊢ (𝐴 ⊆ ω → 𝐴 ⊆ On)
4	3	adantr 486	. 2 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → 𝐴 ⊆ On)
5		frfnom 8428	. . . 4 ⊢ (rec((𝑥 ∈ V ↦ ∩ (𝐴 ∖ suc 𝑥)), ∩ 𝐴) ↾ ω) Fn ω
6		unblem.2	. . . . 5 ⊢ 𝐹 = (rec((𝑥 ∈ V ↦ ∩ (𝐴 ∖ suc 𝑥)), ∩ 𝐴) ↾ ω)
7	6	fneq1i 6636	. . . 4 ⊢ (𝐹 Fn ω ↔ (rec((𝑥 ∈ V ↦ ∩ (𝐴 ∖ suc 𝑥)), ∩ 𝐴) ↾ ω) Fn ω)
8	5, 7	mpbir 234	. . 3 ⊢ 𝐹 Fn ω
9	6	unblem2 9260	. . . 4 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → (𝑧 ∈ ω → (𝐹‘𝑧) ∈ 𝐴))
10	9	ralrimiv 3158	. . 3 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → ∀𝑧 ∈ ω (𝐹‘𝑧) ∈ 𝐴)
11		ffnfv 7118	. . . 4 ⊢ (𝐹:ω⟶𝐴 ↔ (𝐹 Fn ω ∧ ∀𝑧 ∈ ω (𝐹‘𝑧) ∈ 𝐴))
12	11	biimpri 231	. . 3 ⊢ ((𝐹 Fn ω ∧ ∀𝑧 ∈ ω (𝐹‘𝑧) ∈ 𝐴) → 𝐹:ω⟶𝐴)
13	8, 10, 12	sylancr 599	. 2 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → 𝐹:ω⟶𝐴)
14	6	unblem3 9261	. . 3 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → (𝑧 ∈ ω → (𝐹‘𝑧) ∈ (𝐹‘suc 𝑧)))
15	14	ralrimiv 3158	. 2 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → ∀𝑧 ∈ ω (𝐹‘𝑧) ∈ (𝐹‘suc 𝑧))
16		omsmo 8650	. 2 ⊢ (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑧 ∈ ω (𝐹‘𝑧) ∈ (𝐹‘suc 𝑧)) → 𝐹:ω–1-1→𝐴)
17	4, 13, 15, 16	syl21anc 851	1 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → 𝐹:ω–1-1→𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091  Vcvv 3457   ∖ cdif 3903   ⊆ wss 3906  ∩ cint 4914   ↦ cmpt 5194   ↾ cres 5665  Oncon0 6364  suc csuc 6366   Fn wfn 6535  ⟶wf 6536  –1-1→wf1 6537  ‘cfv 6540  ωcom 7868  reccrdg 8402

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-om 7869  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403

This theorem is used by:  unbnn  9263