Theorem f1ocnt 33085

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 33083 or iundisj2cnt 33084. (Contributed by Thierry Arnoux, 25-Jul-2020.)

Assertion

Ref	Expression
f1ocnt	⊢ (𝐴 ≼ ω → ∃𝑓(𝑓:dom 𝑓–1-1-onto→𝐴 ∧ (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 6859	. . . . . . 7 ⊢ ∅:∅–1-1-onto→∅
2		eqidd 2770	. . . . . . . 8 ⊢ (𝐴 = ∅ → ∅ = ∅)
3		dm0 5911	. . . . . . . . 9 ⊢ dom ∅ = ∅
4	3	a1i 11	. . . . . . . 8 ⊢ (𝐴 = ∅ → dom ∅ = ∅)
5		id 23	. . . . . . . 8 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
6	2, 4, 5	f1oeq123d 6815	. . . . . . 7 ⊢ (𝐴 = ∅ → (∅:dom ∅–1-1-onto→𝐴 ↔ ∅:∅–1-1-onto→∅))
7	1, 6	mpbiri 261	. . . . . 6 ⊢ (𝐴 = ∅ → ∅:dom ∅–1-1-onto→𝐴)
8		fveq2 6882	. . . . . . . . . . . . 13 ⊢ (𝐴 = ∅ → (♯‘𝐴) = (♯‘∅))
9		hash0 14402	. . . . . . . . . . . . 13 ⊢ (♯‘∅) = 0
10	8, 9	eqtrdi 2820	. . . . . . . . . . . 12 ⊢ (𝐴 = ∅ → (♯‘𝐴) = 0)
11	10	oveq1d 7426	. . . . . . . . . . 11 ⊢ (𝐴 = ∅ → ((♯‘𝐴) + 1) = (0 + 1))
12		0p1e1 12360	. . . . . . . . . . 11 ⊢ (0 + 1) = 1
13	11, 12	eqtrdi 2820	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → ((♯‘𝐴) + 1) = 1)
14	13	oveq2d 7427	. . . . . . . . 9 ⊢ (𝐴 = ∅ → (1..^((♯‘𝐴) + 1)) = (1..^1))
15		fzo0 13711	. . . . . . . . 9 ⊢ (1..^1) = ∅
16	14, 15	eqtrdi 2820	. . . . . . . 8 ⊢ (𝐴 = ∅ → (1..^((♯‘𝐴) + 1)) = ∅)
17	4, 16	eqtr4d 2807	. . . . . . 7 ⊢ (𝐴 = ∅ → dom ∅ = (1..^((♯‘𝐴) + 1)))
18	17	olcd 887	. . . . . 6 ⊢ (𝐴 = ∅ → (dom ∅ = ℕ ∨ dom ∅ = (1..^((♯‘𝐴) + 1))))
19	7, 18	jca 520	. . . . 5 ⊢ (𝐴 = ∅ → (∅:dom ∅–1-1-onto→𝐴 ∧ (dom ∅ = ℕ ∨ dom ∅ = (1..^((♯‘𝐴) + 1)))))
20		0ex 5272	. . . . . 6 ⊢ ∅ ∈ V
21		id 23	. . . . . . . 8 ⊢ (𝑓 = ∅ → 𝑓 = ∅)
22		dmeq 5894	. . . . . . . 8 ⊢ (𝑓 = ∅ → dom 𝑓 = dom ∅)
23		eqidd 2770	. . . . . . . 8 ⊢ (𝑓 = ∅ → 𝐴 = 𝐴)
24	21, 22, 23	f1oeq123d 6815	. . . . . . 7 ⊢ (𝑓 = ∅ → (𝑓:dom 𝑓–1-1-onto→𝐴 ↔ ∅:dom ∅–1-1-onto→𝐴))
25	22	eqeq1d 2771	. . . . . . . 8 ⊢ (𝑓 = ∅ → (dom 𝑓 = ℕ ↔ dom ∅ = ℕ))
26	22	eqeq1d 2771	. . . . . . . 8 ⊢ (𝑓 = ∅ → (dom 𝑓 = (1..^((♯‘𝐴) + 1)) ↔ dom ∅ = (1..^((♯‘𝐴) + 1))))
27	25, 26	orbi12d 931	. . . . . . 7 ⊢ (𝑓 = ∅ → ((dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1))) ↔ (dom ∅ = ℕ ∨ dom ∅ = (1..^((♯‘𝐴) + 1)))))
28	24, 27	anbi12d 643	. . . . . 6 ⊢ (𝑓 = ∅ → ((𝑓:dom 𝑓–1-1-onto→𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))) ↔ (∅:dom ∅–1-1-onto→𝐴 ∧ (dom ∅ = ℕ ∨ dom ∅ = (1..^((♯‘𝐴) + 1))))))
29	20, 28	spcev 3574	. . . . 5 ⊢ ((∅:dom ∅–1-1-onto→𝐴 ∧ (dom ∅ = ℕ ∨ dom ∅ = (1..^((♯‘𝐴) + 1)))) → ∃𝑓(𝑓:dom 𝑓–1-1-onto→𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))
30	19, 29	syl 18	. . . 4 ⊢ (𝐴 = ∅ → ∃𝑓(𝑓:dom 𝑓–1-1-onto→𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))
31	30	adantl 486	. . 3 ⊢ (((𝐴 ≼ ω ∧ 𝐴 ∈ Fin) ∧ 𝐴 = ∅) → ∃𝑓(𝑓:dom 𝑓–1-1-onto→𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))
32		f1odm 6825	. . . . . . . . . . 11 ⊢ (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → dom 𝑓 = (1...(♯‘𝐴)))
33	32	f1oeq2d 6817	. . . . . . . . . 10 ⊢ (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → (𝑓:dom 𝑓–1-1-onto→𝐴 ↔ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴))
34	33	ibir 271	. . . . . . . . 9 ⊢ (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → 𝑓:dom 𝑓–1-1-onto→𝐴)
35	34	adantl 486	. . . . . . . 8 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → 𝑓:dom 𝑓–1-1-onto→𝐴)
36	32	adantl 486	. . . . . . . . . 10 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → dom 𝑓 = (1...(♯‘𝐴)))
37		simpl 487	. . . . . . . . . . . 12 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → (♯‘𝐴) ∈ ℕ)
38	37	nnzd 12616	. . . . . . . . . . 11 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → (♯‘𝐴) ∈ ℤ)
39		fzval3 13762	. . . . . . . . . . 11 ⊢ ((♯‘𝐴) ∈ ℤ → (1...(♯‘𝐴)) = (1..^((♯‘𝐴) + 1)))
40	38, 39	syl 18	. . . . . . . . . 10 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → (1...(♯‘𝐴)) = (1..^((♯‘𝐴) + 1)))
41	36, 40	eqtrd 2804	. . . . . . . . 9 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → dom 𝑓 = (1..^((♯‘𝐴) + 1)))
42	41	olcd 887	. . . . . . . 8 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1))))
43	35, 42	jca 520	. . . . . . 7 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → (𝑓:dom 𝑓–1-1-onto→𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))
44	43	ex 417	. . . . . 6 ⊢ ((♯‘𝐴) ∈ ℕ → (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → (𝑓:dom 𝑓–1-1-onto→𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1))))))
45	44	eximdv 1944	. . . . 5 ⊢ ((♯‘𝐴) ∈ ℕ → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → ∃𝑓(𝑓:dom 𝑓–1-1-onto→𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1))))))
46	45	imp 411	. . . 4 ⊢ (((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → ∃𝑓(𝑓:dom 𝑓–1-1-onto→𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))
47	46	adantl 486	. . 3 ⊢ (((𝐴 ≼ ω ∧ 𝐴 ∈ Fin) ∧ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ∃𝑓(𝑓:dom 𝑓–1-1-onto→𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))
48		fz1f1o 15760	. . . 4 ⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)))
49	48	adantl 486	. . 3 ⊢ ((𝐴 ≼ ω ∧ 𝐴 ∈ Fin) → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)))
50	31, 47, 49	mpjaodan 973	. 2 ⊢ ((𝐴 ≼ ω ∧ 𝐴 ∈ Fin) → ∃𝑓(𝑓:dom 𝑓–1-1-onto→𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))
51		isfinite 9620	. . . . . . . . . 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 628	. . . . . . 7 ⊢ ((𝐴 ≼ ω ∧ ¬ 𝐴 ∈ Fin) → (𝐴 ≼ ω ∧ ¬ 𝐴 ≺ ω))
55		bren2 8979	. . . . . . 7 ⊢ (𝐴 ≈ ω ↔ (𝐴 ≼ ω ∧ ¬ 𝐴 ≺ ω))
56	54, 55	sylibr 237	. . . . . 6 ⊢ ((𝐴 ≼ ω ∧ ¬ 𝐴 ∈ Fin) → 𝐴 ≈ ω)
57		nnenom 14015	. . . . . . 7 ⊢ ℕ ≈ ω
58	57	ensymi 9000	. . . . . 6 ⊢ ω ≈ ℕ
59		entr 9002	. . . . . 6 ⊢ ((𝐴 ≈ ω ∧ ω ≈ ℕ) → 𝐴 ≈ ℕ)
60	56, 58, 59	sylancl 597	. . . . 5 ⊢ ((𝐴 ≼ ω ∧ ¬ 𝐴 ∈ Fin) → 𝐴 ≈ ℕ)
61		bren 8952	. . . . 5 ⊢ (𝐴 ≈ ℕ ↔ ∃𝑔 𝑔:𝐴–1-1-onto→ℕ)
62	60, 61	sylib 221	. . . 4 ⊢ ((𝐴 ≼ ω ∧ ¬ 𝐴 ∈ Fin) → ∃𝑔 𝑔:𝐴–1-1-onto→ℕ)
63		f1oexbi 7924	. . . 4 ⊢ (∃𝑔 𝑔:𝐴–1-1-onto→ℕ ↔ ∃𝑓 𝑓:ℕ–1-1-onto→𝐴)
64	62, 63	sylib 221	. . 3 ⊢ ((𝐴 ≼ ω ∧ ¬ 𝐴 ∈ Fin) → ∃𝑓 𝑓:ℕ–1-1-onto→𝐴)
65		f1odm 6825	. . . . . . 7 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → dom 𝑓 = ℕ)
66	65	f1oeq2d 6817	. . . . . 6 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → (𝑓:dom 𝑓–1-1-onto→𝐴 ↔ 𝑓:ℕ–1-1-onto→𝐴))
67	66	ibir 271	. . . . 5 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → 𝑓:dom 𝑓–1-1-onto→𝐴)
68	65	orcd 886	. . . . 5 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1))))
69	67, 68	jca 520	. . . 4 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → (𝑓:dom 𝑓–1-1-onto→𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))
70	69	eximi 1862	. . 3 ⊢ (∃𝑓 𝑓:ℕ–1-1-onto→𝐴 → ∃𝑓(𝑓:dom 𝑓–1-1-onto→𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))
71	64, 70	syl 18	. 2 ⊢ ((𝐴 ≼ ω ∧ ¬ 𝐴 ∈ Fin) → ∃𝑓(𝑓:dom 𝑓–1-1-onto→𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))
72	50, 71	pm2.61dan 824	1 ⊢ (𝐴 ≼ ω → ∃𝑓(𝑓:dom 𝑓–1-1-onto→𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∨ wo 860   = wceq 1567  ∃wex 1806   ∈ wcel 2149  ∅c0 4294   class class class wbr 5113  dom cdm 5662  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7411  ωcom 7861   ≈ cen 8939   ≼ cdom 8940   ≺ csdm 8941  Fincfn 8942  0cc0 11099  1c1 11100   + caddc 11102  ℕcn 12232  ℤcz 12590  ...cfz 13534  ..^cfzo 13681  ♯chash 14365

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-pow 5337  ax-pr 5405  ax-un 7733  ax-inf2 9609  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176

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-nel 3071  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-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-er 8693  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-card 9924  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-nn 12233  df-n0 12504  df-z 12591  df-uz 12862  df-fz 13535  df-fzo 13682  df-hash 14366

This theorem is referenced by: (None)