Theorem f1ocnt 33488

Description: Given a countable set 𝐴, number its elements by providing a one-to-one mapping either with ℕ or an integer range starting from 1. The domain of the function can then be used with iundisjcnt 33486 or iundisj2cnt 33487. (Contributed by Thierry Arnoux, 25-Jul-2020.)

Assertion

Ref	Expression
f1ocnt	⊢ (𝐴 ≼ ω → ∃𝑓(𝑓:dom 𝑓⟶1-1onto𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))

Distinct variable group:   𝐴,𝑓

Proof of Theorem f1ocnt

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1o0 6893	. . . . . . 7 ⊢ ∅:∅⟶1-1onto∅
2		eqidd 2764	. . . . . . . 8 ⊢ (𝐴 = ∅ → ∅ = ∅)
3		dm0 5914	. . . . . . . . 9 ⊢ dom ∅ = ∅
4	3	a1i 11	. . . . . . . 8 ⊢ (𝐴 = ∅ → dom ∅ = ∅)
5		id 23	. . . . . . . 8 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
6	2, 4, 5	f1oeq123d 6848	. . . . . . 7 ⊢ (𝐴 = ∅ → (∅:dom ∅⟶1-1onto𝐴 ↔ ∅:∅⟶1-1onto∅))
7	1, 6	mpbiri 261	. . . . . 6 ⊢ (𝐴 = ∅ → ∅:dom ∅⟶1-1onto𝐴)
8		fveq2 6916	. . . . . . . . . . . . 13 ⊢ (𝐴 = ∅ → (♯‘𝐴) = (♯‘∅))
9		hash0 14552	. . . . . . . . . . . . 13 ⊢ (♯‘∅) = 0
10	8, 9	eqtrdi 2814	. . . . . . . . . . . 12 ⊢ (𝐴 = ∅ → (♯‘𝐴) = 0)
11	10	oveq1d 7466	. . . . . . . . . . 11 ⊢ (𝐴 = ∅ → ((♯‘𝐴) + 1) = (0 + 1))
12		0p1e1 12503	. . . . . . . . . . 11 ⊢ (0 + 1) = 1
13	11, 12	eqtrdi 2814	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → ((♯‘𝐴) + 1) = 1)
14	13	oveq2d 7467	. . . . . . . . 9 ⊢ (𝐴 = ∅ → (1..^((♯‘𝐴) + 1)) = (1..^1))
15		fzo0 13858	. . . . . . . . 9 ⊢ (1..^1) = ∅
16	14, 15	eqtrdi 2814	. . . . . . . 8 ⊢ (𝐴 = ∅ → (1..^((♯‘𝐴) + 1)) = ∅)
17	4, 16	eqtr4d 2801	. . . . . . 7 ⊢ (𝐴 = ∅ → dom ∅ = (1..^((♯‘𝐴) + 1)))
18	17	olcd 888	. . . . . 6 ⊢ (𝐴 = ∅ → (dom ∅ = ℕ ∨ dom ∅ = (1..^((♯‘𝐴) + 1))))
19	7, 18	jca 521	. . . . 5 ⊢ (𝐴 = ∅ → (∅:dom ∅⟶1-1onto𝐴 ∧ (dom ∅ = ℕ ∨ dom ∅ = (1..^((♯‘𝐴) + 1)))))
20		0ex 5267	. . . . . 6 ⊢ ∅ ∈ V
21		id 23	. . . . . . . 8 ⊢ (𝑓 = ∅ → 𝑓 = ∅)
22		dmeq 5897	. . . . . . . 8 ⊢ (𝑓 = ∅ → dom 𝑓 = dom ∅)
23		eqidd 2764	. . . . . . . 8 ⊢ (𝑓 = ∅ → 𝐴 = 𝐴)
24	21, 22, 23	f1oeq123d 6848	. . . . . . 7 ⊢ (𝑓 = ∅ → (𝑓:dom 𝑓⟶1-1onto𝐴 ↔ ∅:dom ∅⟶1-1onto𝐴))
25	22	eqeq1d 2765	. . . . . . . 8 ⊢ (𝑓 = ∅ → (dom 𝑓 = ℕ ↔ dom ∅ = ℕ))
26	22	eqeq1d 2765	. . . . . . . 8 ⊢ (𝑓 = ∅ → (dom 𝑓 = (1..^((♯‘𝐴) + 1)) ↔ dom ∅ = (1..^((♯‘𝐴) + 1))))
27	25, 26	orbi12d 932	. . . . . . 7 ⊢ (𝑓 = ∅ → ((dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1))) ↔ (dom ∅ = ℕ ∨ dom ∅ = (1..^((♯‘𝐴) + 1)))))
28	24, 27	anbi12d 644	. . . . . 6 ⊢ (𝑓 = ∅ → ((𝑓:dom 𝑓⟶1-1onto𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))) ↔ (∅:dom ∅⟶1-1onto𝐴 ∧ (dom ∅ = ℕ ∨ dom ∅ = (1..^((♯‘𝐴) + 1))))))
29	20, 28	spcev 3564	. . . . 5 ⊢ ((∅:dom ∅⟶1-1onto𝐴 ∧ (dom ∅ = ℕ ∨ dom ∅ = (1..^((♯‘𝐴) + 1)))) → ∃𝑓(𝑓:dom 𝑓⟶1-1onto𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))
30	19, 29	syl 18	. . . 4 ⊢ (𝐴 = ∅ → ∃𝑓(𝑓:dom 𝑓⟶1-1onto𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))
31	30	adantl 487	. . 3 ⊢ (((𝐴 ≼ ω ∧ 𝐴 ∈ Fin) ∧ 𝐴 = ∅) → ∃𝑓(𝑓:dom 𝑓⟶1-1onto𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))
32		f1odm 6858	. . . . . . . . . . 11 ⊢ (𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴 → dom 𝑓 = (1...(♯‘𝐴)))
33	32	f1oeq2d 6850	. . . . . . . . . 10 ⊢ (𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴 → (𝑓:dom 𝑓⟶1-1onto𝐴 ↔ 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴))
34	33	ibir 271	. . . . . . . . 9 ⊢ (𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴 → 𝑓:dom 𝑓⟶1-1onto𝐴)
35	34	adantl 487	. . . . . . . 8 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴) → 𝑓:dom 𝑓⟶1-1onto𝐴)
36	32	adantl 487	. . . . . . . . . 10 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴) → dom 𝑓 = (1...(♯‘𝐴)))
37		simpl 488	. . . . . . . . . . . 12 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴) → (♯‘𝐴) ∈ ℕ)
38	37	nnzd 12759	. . . . . . . . . . 11 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴) → (♯‘𝐴) ∈ ℤ)
39		fzval3 13909	. . . . . . . . . . 11 ⊢ ((♯‘𝐴) ∈ ℤ → (1...(♯‘𝐴)) = (1..^((♯‘𝐴) + 1)))
40	38, 39	syl 18	. . . . . . . . . 10 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴) → (1...(♯‘𝐴)) = (1..^((♯‘𝐴) + 1)))
41	36, 40	eqtrd 2798	. . . . . . . . 9 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴) → dom 𝑓 = (1..^((♯‘𝐴) + 1)))
42	41	olcd 888	. . . . . . . 8 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴) → (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1))))
43	35, 42	jca 521	. . . . . . 7 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴) → (𝑓:dom 𝑓⟶1-1onto𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))
44	43	ex 418	. . . . . 6 ⊢ ((♯‘𝐴) ∈ ℕ → (𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴 → (𝑓:dom 𝑓⟶1-1onto𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1))))))
45	44	eximdv 1952	. . . . 5 ⊢ ((♯‘𝐴) ∈ ℕ → (∃𝑓 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴 → ∃𝑓(𝑓:dom 𝑓⟶1-1onto𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1))))))
46	45	imp 412	. . . 4 ⊢ (((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴) → ∃𝑓(𝑓:dom 𝑓⟶1-1onto𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))
47	46	adantl 487	. . 3 ⊢ (((𝐴 ≼ ω ∧ 𝐴 ∈ Fin) ∧ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴)) → ∃𝑓(𝑓:dom 𝑓⟶1-1onto𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))
48		fz1f1o 15920	. . . 4 ⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴)))
49	48	adantl 487	. . 3 ⊢ ((𝐴 ≼ ω ∧ 𝐴 ∈ Fin) → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴)))
50	31, 47, 49	mpjaodan 973	. 2 ⊢ ((𝐴 ≼ ω ∧ 𝐴 ∈ Fin) → ∃𝑓(𝑓:dom 𝑓⟶1-1onto𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))
51		isfinite 9690	. . . . . . . . . 10 ⊢ (𝐴 ∈ Fin ↔ 𝐴 ≺ ω)
52	51	notbii 323	. . . . . . . . 9 ⊢ (¬ 𝐴 ∈ Fin ↔ ¬ 𝐴 ≺ ω)
53	52	biimpi 219	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ Fin → ¬ 𝐴 ≺ ω)
54	53	anim2i 629	. . . . . . 7 ⊢ ((𝐴 ≼ ω ∧ ¬ 𝐴 ∈ Fin) → (𝐴 ≼ ω ∧ ¬ 𝐴 ≺ ω))
55		bren2 9048	. . . . . . 7 ⊢ (𝐴 ≈ ω ↔ (𝐴 ≼ ω ∧ ¬ 𝐴 ≺ ω))
56	54, 55	sylibr 237	. . . . . 6 ⊢ ((𝐴 ≼ ω ∧ ¬ 𝐴 ∈ Fin) → 𝐴 ≈ ω)
57		nnenom 14163	. . . . . . 7 ⊢ ℕ ≈ ω
58	57	ensymi 9069	. . . . . 6 ⊢ ω ≈ ℕ
59		entr 9071	. . . . . 6 ⊢ ((𝐴 ≈ ω ∧ ω ≈ ℕ) → 𝐴 ≈ ℕ)
60	56, 58, 59	sylancl 598	. . . . 5 ⊢ ((𝐴 ≼ ω ∧ ¬ 𝐴 ∈ Fin) → 𝐴 ≈ ℕ)
61		bren 9021	. . . . 5 ⊢ (𝐴 ≈ ℕ ↔ ∃𝑔 𝑔:𝐴⟶1-1ontoℕ)
62	60, 61	sylib 221	. . . 4 ⊢ ((𝐴 ≼ ω ∧ ¬ 𝐴 ∈ Fin) → ∃𝑔 𝑔:𝐴⟶1-1ontoℕ)
63		f1oexbi 7974	. . . 4 ⊢ (∃𝑔 𝑔:𝐴⟶1-1ontoℕ ↔ ∃𝑓 𝑓:ℕ⟶1-1onto𝐴)
64	62, 63	sylib 221	. . 3 ⊢ ((𝐴 ≼ ω ∧ ¬ 𝐴 ∈ Fin) → ∃𝑓 𝑓:ℕ⟶1-1onto𝐴)
65		f1odm 6858	. . . . . . 7 ⊢ (𝑓:ℕ⟶1-1onto𝐴 → dom 𝑓 = ℕ)
66	65	f1oeq2d 6850	. . . . . 6 ⊢ (𝑓:ℕ⟶1-1onto𝐴 → (𝑓:dom 𝑓⟶1-1onto𝐴 ↔ 𝑓:ℕ⟶1-1onto𝐴))
67	66	ibir 271	. . . . 5 ⊢ (𝑓:ℕ⟶1-1onto𝐴 → 𝑓:dom 𝑓⟶1-1onto𝐴)
68	65	orcd 887	. . . . 5 ⊢ (𝑓:ℕ⟶1-1onto𝐴 → (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1))))
69	67, 68	jca 521	. . . 4 ⊢ (𝑓:ℕ⟶1-1onto𝐴 → (𝑓:dom 𝑓⟶1-1onto𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))
70	69	eximi 1868	. . 3 ⊢ (∃𝑓 𝑓:ℕ⟶1-1onto𝐴 → ∃𝑓(𝑓:dom 𝑓⟶1-1onto𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))
71	64, 70	syl 18	. 2 ⊢ ((𝐴 ≼ ω ∧ ¬ 𝐴 ∈ Fin) → ∃𝑓(𝑓:dom 𝑓⟶1-1onto𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))
72	50, 71	pm2.61dan 825	1 ⊢ (𝐴 ≼ ω → ∃𝑓(𝑓:dom 𝑓⟶1-1onto𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∅c0 4281   class class class wbr 5108  dom cdm 5658  ⟶1-1ontowf1o 6565  ‘cfv 6566  (class class class)co 7451  ωcom 7911   ≈ cen 9008   ≼ cdom 9009   ≺ csdm 9010  Fincfn 9011  0cc0 11239  1c1 11240   + caddc 11242  ℕcn 12375  ℤcz 12733  ...cfz 13679  ..^cfzo 13828  ♯chash 14515

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-inf2 9679  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-er 8755  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-card 10057  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-n0 12647  df-z 12734  df-uz 13006  df-fz 13680  df-fzo 13829  df-hash 14516

This theorem is used by: (None)