Theorem f1ocnt 32880

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 32878 or iundisj2cnt 32879. (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 6811	. . . . . . 7 ⊢ ∅:∅–1-1-onto→∅
2		eqidd 2737	. . . . . . . 8 ⊢ (𝐴 = ∅ → ∅ = ∅)
3		dm0 5869	. . . . . . . . 9 ⊢ dom ∅ = ∅
4	3	a1i 11	. . . . . . . 8 ⊢ (𝐴 = ∅ → dom ∅ = ∅)
5		id 22	. . . . . . . 8 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
6	2, 4, 5	f1oeq123d 6768	. . . . . . 7 ⊢ (𝐴 = ∅ → (∅:dom ∅–1-1-onto→𝐴 ↔ ∅:∅–1-1-onto→∅))
7	1, 6	mpbiri 258	. . . . . 6 ⊢ (𝐴 = ∅ → ∅:dom ∅–1-1-onto→𝐴)
8		fveq2 6834	. . . . . . . . . . . . 13 ⊢ (𝐴 = ∅ → (♯‘𝐴) = (♯‘∅))
9		hash0 14290	. . . . . . . . . . . . 13 ⊢ (♯‘∅) = 0
10	8, 9	eqtrdi 2787	. . . . . . . . . . . 12 ⊢ (𝐴 = ∅ → (♯‘𝐴) = 0)
11	10	oveq1d 7373	. . . . . . . . . . 11 ⊢ (𝐴 = ∅ → ((♯‘𝐴) + 1) = (0 + 1))
12		0p1e1 12262	. . . . . . . . . . 11 ⊢ (0 + 1) = 1
13	11, 12	eqtrdi 2787	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → ((♯‘𝐴) + 1) = 1)
14	13	oveq2d 7374	. . . . . . . . 9 ⊢ (𝐴 = ∅ → (1..^((♯‘𝐴) + 1)) = (1..^1))
15		fzo0 13599	. . . . . . . . 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 874	. . . . . 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 5252	. . . . . 6 ⊢ ∅ ∈ V
21		id 22	. . . . . . . 8 ⊢ (𝑓 = ∅ → 𝑓 = ∅)
22		dmeq 5852	. . . . . . . 8 ⊢ (𝑓 = ∅ → dom 𝑓 = dom ∅)
23		eqidd 2737	. . . . . . . 8 ⊢ (𝑓 = ∅ → 𝐴 = 𝐴)
24	21, 22, 23	f1oeq123d 6768	. . . . . . 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 918	. . . . . . 7 ⊢ (𝑓 = ∅ → ((dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1))) ↔ (dom ∅ = ℕ ∨ dom ∅ = (1..^((♯‘𝐴) + 1)))))
28	24, 27	anbi12d 632	. . . . . 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 3560	. . . . 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 6778	. . . . . . . . . . 11 ⊢ (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → dom 𝑓 = (1...(♯‘𝐴)))
33	32	f1oeq2d 6770	. . . . . . . . . 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 12514	. . . . . . . . . . 11 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → (♯‘𝐴) ∈ ℤ)
39		fzval3 13650	. . . . . . . . . . 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 874	. . . . . . . 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 1918	. . . . 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 15633	. . . 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 960	. 2 ⊢ ((𝐴 ≼ ω ∧ 𝐴 ∈ Fin) → ∃𝑓(𝑓:dom 𝑓–1-1-onto→𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))
51		isfinite 9561	. . . . . . . . . 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 617	. . . . . . 7 ⊢ ((𝐴 ≼ ω ∧ ¬ 𝐴 ∈ Fin) → (𝐴 ≼ ω ∧ ¬ 𝐴 ≺ ω))
55		bren2 8920	. . . . . . 7 ⊢ (𝐴 ≈ ω ↔ (𝐴 ≼ ω ∧ ¬ 𝐴 ≺ ω))
56	54, 55	sylibr 234	. . . . . 6 ⊢ ((𝐴 ≼ ω ∧ ¬ 𝐴 ∈ Fin) → 𝐴 ≈ ω)
57		nnenom 13903	. . . . . . 7 ⊢ ℕ ≈ ω
58	57	ensymi 8941	. . . . . 6 ⊢ ω ≈ ℕ
59		entr 8943	. . . . . 6 ⊢ ((𝐴 ≈ ω ∧ ω ≈ ℕ) → 𝐴 ≈ ℕ)
60	56, 58, 59	sylancl 586	. . . . 5 ⊢ ((𝐴 ≼ ω ∧ ¬ 𝐴 ∈ Fin) → 𝐴 ≈ ℕ)
61		bren 8893	. . . . 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 7870	. . . 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 6778	. . . . . . 7 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → dom 𝑓 = ℕ)
66	65	f1oeq2d 6770	. . . . . 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 873	. . . . 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 1836	. . 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 812	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 847   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∅c0 4285   class class class wbr 5098  dom cdm 5624  –1-1-onto→wf1o 6491  ‘cfv 6492  (class class class)co 7358  ωcom 7808   ≈ cen 8880   ≼ cdom 8881   ≺ csdm 8882  Fincfn 8883  0cc0 11026  1c1 11027   + caddc 11029  ℕcn 12145  ℤcz 12488  ...cfz 13423  ..^cfzo 13570  ♯chash 14253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-inf2 9550  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  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 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-n0 12402  df-z 12489  df-uz 12752  df-fz 13424  df-fzo 13571  df-hash 14254

This theorem is referenced by: (None)