Theorem f1ocnt 29559

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 29557 or iundisj2cnt 29558. (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 6173	. . . . . . 7 ⊢ ∅:∅–1-1-onto→∅
2		eqidd 2623	. . . . . . . 8 ⊢ (𝐴 = ∅ → ∅ = ∅)
3		dm0 5339	. . . . . . . . 9 ⊢ dom ∅ = ∅
4	3	a1i 11	. . . . . . . 8 ⊢ (𝐴 = ∅ → dom ∅ = ∅)
5		id 22	. . . . . . . 8 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
6	2, 4, 5	f1oeq123d 6133	. . . . . . 7 ⊢ (𝐴 = ∅ → (∅:dom ∅–1-1-onto→𝐴 ↔ ∅:∅–1-1-onto→∅))
7	1, 6	mpbiri 248	. . . . . 6 ⊢ (𝐴 = ∅ → ∅:dom ∅–1-1-onto→𝐴)
8		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝐴 = ∅ → (#‘𝐴) = (#‘∅))
9		hash0 13158	. . . . . . . . . . . . 13 ⊢ (#‘∅) = 0
10	8, 9	syl6eq 2672	. . . . . . . . . . . 12 ⊢ (𝐴 = ∅ → (#‘𝐴) = 0)
11	10	oveq1d 6665	. . . . . . . . . . 11 ⊢ (𝐴 = ∅ → ((#‘𝐴) + 1) = (0 + 1))
12		0p1e1 11132	. . . . . . . . . . 11 ⊢ (0 + 1) = 1
13	11, 12	syl6eq 2672	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → ((#‘𝐴) + 1) = 1)
14	13	oveq2d 6666	. . . . . . . . 9 ⊢ (𝐴 = ∅ → (1..^((#‘𝐴) + 1)) = (1..^1))
15		fzo0 12492	. . . . . . . . 9 ⊢ (1..^1) = ∅
16	14, 15	syl6eq 2672	. . . . . . . 8 ⊢ (𝐴 = ∅ → (1..^((#‘𝐴) + 1)) = ∅)
17	4, 16	eqtr4d 2659	. . . . . . 7 ⊢ (𝐴 = ∅ → dom ∅ = (1..^((#‘𝐴) + 1)))
18	17	olcd 408	. . . . . 6 ⊢ (𝐴 = ∅ → (dom ∅ = ℕ ∨ dom ∅ = (1..^((#‘𝐴) + 1))))
19	7, 18	jca 554	. . . . 5 ⊢ (𝐴 = ∅ → (∅:dom ∅–1-1-onto→𝐴 ∧ (dom ∅ = ℕ ∨ dom ∅ = (1..^((#‘𝐴) + 1)))))
20		0ex 4790	. . . . . 6 ⊢ ∅ ∈ V
21		id 22	. . . . . . . 8 ⊢ (𝑓 = ∅ → 𝑓 = ∅)
22		dmeq 5324	. . . . . . . 8 ⊢ (𝑓 = ∅ → dom 𝑓 = dom ∅)
23		eqidd 2623	. . . . . . . 8 ⊢ (𝑓 = ∅ → 𝐴 = 𝐴)
24	21, 22, 23	f1oeq123d 6133	. . . . . . 7 ⊢ (𝑓 = ∅ → (𝑓:dom 𝑓–1-1-onto→𝐴 ↔ ∅:dom ∅–1-1-onto→𝐴))
25	22	eqeq1d 2624	. . . . . . . 8 ⊢ (𝑓 = ∅ → (dom 𝑓 = ℕ ↔ dom ∅ = ℕ))
26	22	eqeq1d 2624	. . . . . . . 8 ⊢ (𝑓 = ∅ → (dom 𝑓 = (1..^((#‘𝐴) + 1)) ↔ dom ∅ = (1..^((#‘𝐴) + 1))))
27	25, 26	orbi12d 746	. . . . . . 7 ⊢ (𝑓 = ∅ → ((dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((#‘𝐴) + 1))) ↔ (dom ∅ = ℕ ∨ dom ∅ = (1..^((#‘𝐴) + 1)))))
28	24, 27	anbi12d 747	. . . . . 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 3300	. . . . 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 482	. . 3 ⊢ (((𝐴 ≼ ω ∧ 𝐴 ∈ Fin) ∧ 𝐴 = ∅) → ∃𝑓(𝑓:dom 𝑓–1-1-onto→𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((#‘𝐴) + 1)))))
32		f1odm 6141	. . . . . . . . . . 11 ⊢ (𝑓:(1...(#‘𝐴))–1-1-onto→𝐴 → dom 𝑓 = (1...(#‘𝐴)))
33		f1oeq2 6128	. . . . . . . . . . 11 ⊢ (dom 𝑓 = (1...(#‘𝐴)) → (𝑓:dom 𝑓–1-1-onto→𝐴 ↔ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴))
34	32, 33	syl 17	. . . . . . . . . 10 ⊢ (𝑓:(1...(#‘𝐴))–1-1-onto→𝐴 → (𝑓:dom 𝑓–1-1-onto→𝐴 ↔ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴))
35	34	ibir 257	. . . . . . . . 9 ⊢ (𝑓:(1...(#‘𝐴))–1-1-onto→𝐴 → 𝑓:dom 𝑓–1-1-onto→𝐴)
36	35	adantl 482	. . . . . . . 8 ⊢ (((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) → 𝑓:dom 𝑓–1-1-onto→𝐴)
37	32	adantl 482	. . . . . . . . . 10 ⊢ (((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) → dom 𝑓 = (1...(#‘𝐴)))
38		simpl 473	. . . . . . . . . . . 12 ⊢ (((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) → (#‘𝐴) ∈ ℕ)
39	38	nnzd 11481	. . . . . . . . . . 11 ⊢ (((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) → (#‘𝐴) ∈ ℤ)
40		fzval3 12536	. . . . . . . . . . 11 ⊢ ((#‘𝐴) ∈ ℤ → (1...(#‘𝐴)) = (1..^((#‘𝐴) + 1)))
41	39, 40	syl 17	. . . . . . . . . 10 ⊢ (((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) → (1...(#‘𝐴)) = (1..^((#‘𝐴) + 1)))
42	37, 41	eqtrd 2656	. . . . . . . . 9 ⊢ (((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) → dom 𝑓 = (1..^((#‘𝐴) + 1)))
43	42	olcd 408	. . . . . . . 8 ⊢ (((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) → (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((#‘𝐴) + 1))))
44	36, 43	jca 554	. . . . . . 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)))))
45	44	ex 450	. . . . . 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))))))
46	45	eximdv 1846	. . . . 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))))))
47	46	imp 445	. . . 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)))))
48	47	adantl 482	. . 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)))))
49		fz1f1o 14441	. . . 4 ⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((#‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴)))
50	49	adantl 482	. . 3 ⊢ ((𝐴 ≼ ω ∧ 𝐴 ∈ Fin) → (𝐴 = ∅ ∨ ((#‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴)))
51	31, 48, 50	mpjaodan 827	. 2 ⊢ ((𝐴 ≼ ω ∧ 𝐴 ∈ Fin) → ∃𝑓(𝑓:dom 𝑓–1-1-onto→𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((#‘𝐴) + 1)))))
52		isfinite 8549	. . . . . . . . . 10 ⊢ (𝐴 ∈ Fin ↔ 𝐴 ≺ ω)
53	52	notbii 310	. . . . . . . . 9 ⊢ (¬ 𝐴 ∈ Fin ↔ ¬ 𝐴 ≺ ω)
54	53	biimpi 206	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ Fin → ¬ 𝐴 ≺ ω)
55	54	anim2i 593	. . . . . . 7 ⊢ ((𝐴 ≼ ω ∧ ¬ 𝐴 ∈ Fin) → (𝐴 ≼ ω ∧ ¬ 𝐴 ≺ ω))
56		bren2 7986	. . . . . . 7 ⊢ (𝐴 ≈ ω ↔ (𝐴 ≼ ω ∧ ¬ 𝐴 ≺ ω))
57	55, 56	sylibr 224	. . . . . 6 ⊢ ((𝐴 ≼ ω ∧ ¬ 𝐴 ∈ Fin) → 𝐴 ≈ ω)
58		nnenom 12779	. . . . . . 7 ⊢ ℕ ≈ ω
59	58	ensymi 8006	. . . . . 6 ⊢ ω ≈ ℕ
60		entr 8008	. . . . . 6 ⊢ ((𝐴 ≈ ω ∧ ω ≈ ℕ) → 𝐴 ≈ ℕ)
61	57, 59, 60	sylancl 694	. . . . 5 ⊢ ((𝐴 ≼ ω ∧ ¬ 𝐴 ∈ Fin) → 𝐴 ≈ ℕ)
62		bren 7964	. . . . 5 ⊢ (𝐴 ≈ ℕ ↔ ∃𝑔 𝑔:𝐴–1-1-onto→ℕ)
63	61, 62	sylib 208	. . . 4 ⊢ ((𝐴 ≼ ω ∧ ¬ 𝐴 ∈ Fin) → ∃𝑔 𝑔:𝐴–1-1-onto→ℕ)
64		f1oexbi 7116	. . . 4 ⊢ (∃𝑔 𝑔:𝐴–1-1-onto→ℕ ↔ ∃𝑓 𝑓:ℕ–1-1-onto→𝐴)
65	63, 64	sylib 208	. . 3 ⊢ ((𝐴 ≼ ω ∧ ¬ 𝐴 ∈ Fin) → ∃𝑓 𝑓:ℕ–1-1-onto→𝐴)
66		f1odm 6141	. . . . . . 7 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → dom 𝑓 = ℕ)
67		f1oeq2 6128	. . . . . . 7 ⊢ (dom 𝑓 = ℕ → (𝑓:dom 𝑓–1-1-onto→𝐴 ↔ 𝑓:ℕ–1-1-onto→𝐴))
68	66, 67	syl 17	. . . . . 6 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → (𝑓:dom 𝑓–1-1-onto→𝐴 ↔ 𝑓:ℕ–1-1-onto→𝐴))
69	68	ibir 257	. . . . 5 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → 𝑓:dom 𝑓–1-1-onto→𝐴)
70	66	orcd 407	. . . . 5 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((#‘𝐴) + 1))))
71	69, 70	jca 554	. . . 4 ⊢ (𝑓:ℕ–1-1-onto→𝐴 → (𝑓:dom 𝑓–1-1-onto→𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((#‘𝐴) + 1)))))
72	71	eximi 1762	. . 3 ⊢ (∃𝑓 𝑓:ℕ–1-1-onto→𝐴 → ∃𝑓(𝑓:dom 𝑓–1-1-onto→𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((#‘𝐴) + 1)))))
73	65, 72	syl 17	. 2 ⊢ ((𝐴 ≼ ω ∧ ¬ 𝐴 ∈ Fin) → ∃𝑓(𝑓:dom 𝑓–1-1-onto→𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((#‘𝐴) + 1)))))
74	51, 73	pm2.61dan 832	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   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∅c0 3915   class class class wbr 4653  dom cdm 5114  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650  ωcom 7065   ≈ cen 7952   ≼ cdom 7953   ≺ csdm 7954  Fincfn 7955  0cc0 9936  1c1 9937   + caddc 9939  ℕcn 11020  ℤcz 11377  ...cfz 12326  ..^cfzo 12465  #chash 13117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118

This theorem is referenced by: (None)