Theorem f1ocnt 32873

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 32871 or iundisj2cnt 32872. (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 6817	. . . . . . 7 ⊢ ∅:∅–1-1-onto→∅
2		eqidd 2737	. . . . . . . 8 ⊢ (𝐴 = ∅ → ∅ = ∅)
3		dm0 5875	. . . . . . . . 9 ⊢ dom ∅ = ∅
4	3	a1i 11	. . . . . . . 8 ⊢ (𝐴 = ∅ → dom ∅ = ∅)
5		id 22	. . . . . . . 8 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
6	2, 4, 5	f1oeq123d 6774	. . . . . . 7 ⊢ (𝐴 = ∅ → (∅:dom ∅–1-1-onto→𝐴 ↔ ∅:∅–1-1-onto→∅))
7	1, 6	mpbiri 258	. . . . . 6 ⊢ (𝐴 = ∅ → ∅:dom ∅–1-1-onto→𝐴)
8		fveq2 6840	. . . . . . . . . . . . 13 ⊢ (𝐴 = ∅ → (♯‘𝐴) = (♯‘∅))
9		hash0 14329	. . . . . . . . . . . . 13 ⊢ (♯‘∅) = 0
10	8, 9	eqtrdi 2787	. . . . . . . . . . . 12 ⊢ (𝐴 = ∅ → (♯‘𝐴) = 0)
11	10	oveq1d 7382	. . . . . . . . . . 11 ⊢ (𝐴 = ∅ → ((♯‘𝐴) + 1) = (0 + 1))
12		0p1e1 12298	. . . . . . . . . . 11 ⊢ (0 + 1) = 1
13	11, 12	eqtrdi 2787	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → ((♯‘𝐴) + 1) = 1)
14	13	oveq2d 7383	. . . . . . . . 9 ⊢ (𝐴 = ∅ → (1..^((♯‘𝐴) + 1)) = (1..^1))
15		fzo0 13638	. . . . . . . . 9 ⊢ (1..^1) = ∅
16	14, 15	eqtrdi 2787	. . . . . . . 8 ⊢ (𝐴 = ∅ → (1..^((♯‘𝐴) + 1)) = ∅)
17	4, 16	eqtr4d 2774	. . . . . . 7 ⊢ (𝐴 = ∅ → dom ∅ = (1..^((♯‘𝐴) + 1)))
18	17	olcd 875	. . . . . 6 ⊢ (𝐴 = ∅ → (dom ∅ = ℕ ∨ dom ∅ = (1..^((♯‘𝐴) + 1))))
19	7, 18	jca 511	. . . . 5 ⊢ (𝐴 = ∅ → (∅:dom ∅–1-1-onto→𝐴 ∧ (dom ∅ = ℕ ∨ dom ∅ = (1..^((♯‘𝐴) + 1)))))
20		0ex 5242	. . . . . 6 ⊢ ∅ ∈ V
21		id 22	. . . . . . . 8 ⊢ (𝑓 = ∅ → 𝑓 = ∅)
22		dmeq 5858	. . . . . . . 8 ⊢ (𝑓 = ∅ → dom 𝑓 = dom ∅)
23		eqidd 2737	. . . . . . . 8 ⊢ (𝑓 = ∅ → 𝐴 = 𝐴)
24	21, 22, 23	f1oeq123d 6774	. . . . . . 7 ⊢ (𝑓 = ∅ → (𝑓:dom 𝑓–1-1-onto→𝐴 ↔ ∅:dom ∅–1-1-onto→𝐴))
25	22	eqeq1d 2738	. . . . . . . 8 ⊢ (𝑓 = ∅ → (dom 𝑓 = ℕ ↔ dom ∅ = ℕ))
26	22	eqeq1d 2738	. . . . . . . 8 ⊢ (𝑓 = ∅ → (dom 𝑓 = (1..^((♯‘𝐴) + 1)) ↔ dom ∅ = (1..^((♯‘𝐴) + 1))))
27	25, 26	orbi12d 919	. . . . . . 7 ⊢ (𝑓 = ∅ → ((dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1))) ↔ (dom ∅ = ℕ ∨ dom ∅ = (1..^((♯‘𝐴) + 1)))))
28	24, 27	anbi12d 633	. . . . . 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 3548	. . . . 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 17	. . . 4 ⊢ (𝐴 = ∅ → ∃𝑓(𝑓:dom 𝑓–1-1-onto→𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))
31	30	adantl 481	. . 3 ⊢ (((𝐴 ≼ ω ∧ 𝐴 ∈ Fin) ∧ 𝐴 = ∅) → ∃𝑓(𝑓:dom 𝑓–1-1-onto→𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))
32		f1odm 6784	. . . . . . . . . . 11 ⊢ (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → dom 𝑓 = (1...(♯‘𝐴)))
33	32	f1oeq2d 6776	. . . . . . . . . 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 268	. . . . . . . . 9 ⊢ (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → 𝑓:dom 𝑓–1-1-onto→𝐴)
35	34	adantl 481	. . . . . . . 8 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → 𝑓:dom 𝑓–1-1-onto→𝐴)
36	32	adantl 481	. . . . . . . . . 10 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → dom 𝑓 = (1...(♯‘𝐴)))
37		simpl 482	. . . . . . . . . . . 12 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → (♯‘𝐴) ∈ ℕ)
38	37	nnzd 12550	. . . . . . . . . . 11 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → (♯‘𝐴) ∈ ℤ)
39		fzval3 13689	. . . . . . . . . . 11 ⊢ ((♯‘𝐴) ∈ ℤ → (1...(♯‘𝐴)) = (1..^((♯‘𝐴) + 1)))
40	38, 39	syl 17	. . . . . . . . . 10 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → (1...(♯‘𝐴)) = (1..^((♯‘𝐴) + 1)))
41	36, 40	eqtrd 2771	. . . . . . . . 9 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → dom 𝑓 = (1..^((♯‘𝐴) + 1)))
42	41	olcd 875	. . . . . . . 8 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1))))
43	35, 42	jca 511	. . . . . . 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 412	. . . . . 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 1919	. . . . 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 406	. . . 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 481	. . 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 15672	. . . 4 ⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)))
49	48	adantl 481	. . 3 ⊢ ((𝐴 ≼ ω ∧ 𝐴 ∈ Fin) → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)))
50	31, 47, 49	mpjaodan 961	. 2 ⊢ ((𝐴 ≼ ω ∧ 𝐴 ∈ Fin) → ∃𝑓(𝑓:dom 𝑓–1-1-onto→𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))
51		isfinite 9573	. . . . . . . . . 10 ⊢ (𝐴 ∈ Fin ↔ 𝐴 ≺ ω)
52	51	notbii 320	. . . . . . . . 9 ⊢ (¬ 𝐴 ∈ Fin ↔ ¬ 𝐴 ≺ ω)
53	52	biimpi 216	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ Fin → ¬ 𝐴 ≺ ω)
54	53	anim2i 618	. . . . . . 7 ⊢ ((𝐴 ≼ ω ∧ ¬ 𝐴 ∈ Fin) → (𝐴 ≼ ω ∧ ¬ 𝐴 ≺ ω))
55		bren2 8930	. . . . . . 7 ⊢ (𝐴 ≈ ω ↔ (𝐴 ≼ ω ∧ ¬ 𝐴 ≺ ω))
56	54, 55	sylibr 234	. . . . . 6 ⊢ ((𝐴 ≼ ω ∧ ¬ 𝐴 ∈ Fin) → 𝐴 ≈ ω)
57		nnenom 13942	. . . . . . 7 ⊢ ℕ ≈ ω
58	57	ensymi 8951	. . . . . 6 ⊢ ω ≈ ℕ
59		entr 8953	. . . . . 6 ⊢ ((𝐴 ≈ ω ∧ ω ≈ ℕ) → 𝐴 ≈ ℕ)
60	56, 58, 59	sylancl 587	. . . . 5 ⊢ ((𝐴 ≼ ω ∧ ¬ 𝐴 ∈ Fin) → 𝐴 ≈ ℕ)
61		bren 8903	. . . . 5 ⊢ (𝐴 ≈ ℕ ↔ ∃𝑔 𝑔:𝐴–1-1-onto→ℕ)
62	60, 61	sylib 218	. . . 4 ⊢ ((𝐴 ≼ ω ∧ ¬ 𝐴 ∈ Fin) → ∃𝑔 𝑔:𝐴–1-1-onto→ℕ)
63		f1oexbi 7879	. . . 4 ⊢ (∃𝑔 𝑔:𝐴–1-1-onto→ℕ ↔ ∃𝑓 𝑓:ℕ–1-1-onto→𝐴)
64	62, 63	sylib 218	. . 3 ⊢ ((𝐴 ≼ ω ∧ ¬ 𝐴 ∈ Fin) → ∃𝑓 𝑓:ℕ–1-1-onto→𝐴)
65		f1odm 6784	. . . . . . 7 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → dom 𝑓 = ℕ)
66	65	f1oeq2d 6776	. . . . . 6 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → (𝑓:dom 𝑓–1-1-onto→𝐴 ↔ 𝑓:ℕ–1-1-onto→𝐴))
67	66	ibir 268	. . . . 5 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → 𝑓:dom 𝑓–1-1-onto→𝐴)
68	65	orcd 874	. . . . 5 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1))))
69	67, 68	jca 511	. . . 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 1837	. . 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 17	. 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 813	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 395   ∨ wo 848   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∅c0 4273   class class class wbr 5085  dom cdm 5631  –1-1-onto→wf1o 6497  ‘cfv 6498  (class class class)co 7367  ωcom 7817   ≈ cen 8890   ≼ cdom 8891   ≺ csdm 8892  Fincfn 8893  0cc0 11038  1c1 11039   + caddc 11041  ℕcn 12174  ℤcz 12524  ...cfz 13461  ..^cfzo 13608  ♯chash 14292

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-inf2 9562  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-n0 12438  df-z 12525  df-uz 12789  df-fz 13462  df-fzo 13609  df-hash 14293

This theorem is referenced by: (None)