Theorem unblem4 9255

Description: Lemma for unbnn 9256. 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 7866	. . . 4 ⊢ ω ⊆ On
2		sstr 3953	. . . 4 ⊢ ((𝐴 ⊆ ω ∧ ω ⊆ On) → 𝐴 ⊆ On)
3	1, 2	mpan2 703	. . 3 ⊢ (𝐴 ⊆ ω → 𝐴 ⊆ On)
4	3	adantr 485	. 2 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → 𝐴 ⊆ On)
5		frfnom 8422	. . . 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 6633	. . . 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 9253	. . . 4 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → (𝑧 ∈ ω → (𝐹‘𝑧) ∈ 𝐴))
10	9	ralrimiv 3162	. . 3 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → ∀𝑧 ∈ ω (𝐹‘𝑧) ∈ 𝐴)
11		ffnfv 7115	. . . 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 598	. 2 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → 𝐹:ω⟶𝐴)
14	6	unblem3 9254	. . 3 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → (𝑧 ∈ ω → (𝐹‘𝑧) ∈ (𝐹‘suc 𝑧)))
15	14	ralrimiv 3162	. 2 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → ∀𝑧 ∈ ω (𝐹‘𝑧) ∈ (𝐹‘suc 𝑧))
16		omsmo 8644	. 2 ⊢ (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑧 ∈ ω (𝐹‘𝑧) ∈ (𝐹‘suc 𝑧)) → 𝐹:ω–1-1→𝐴)
17	4, 13, 15, 16	syl21anc 850	1 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → 𝐹:ω–1-1→𝐴)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095  Vcvv 3463   ∖ cdif 3910   ⊆ wss 3913  ∩ cint 4916   ↦ cmpt 5196   ↾ cres 5664  Oncon0 6361  suc csuc 6363   Fn wfn 6532  ⟶wf 6533  –1-1→wf1 6534  ‘cfv 6537  ωcom 7862  reccrdg 8396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-om 7863  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397

This theorem is referenced by:  unbnn  9256