Theorem rp-isfinite5 37863

Description: A set is said to be finite if it can be put in one-to-one correspondence with all the natural numbers between 1 and some 𝑛 ∈ ℕ₀. (Contributed by Richard Penner, 3-Mar-2020.)

Assertion

Ref	Expression
rp-isfinite5	⊢ (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ℕ0 (1...𝑛) ≈ 𝐴)

Distinct variable group:   𝐴,𝑛

Proof of Theorem rp-isfinite5

Step	Hyp	Ref	Expression
1		fvex 6201	. . . 4 ⊢ (#‘𝐴) ∈ V
2		hashcl 13147	. . . . 5 ⊢ (𝐴 ∈ Fin → (#‘𝐴) ∈ ℕ0)
3		isfinite4 13153	. . . . . 6 ⊢ (𝐴 ∈ Fin ↔ (1...(#‘𝐴)) ≈ 𝐴)
4	3	biimpi 206	. . . . 5 ⊢ (𝐴 ∈ Fin → (1...(#‘𝐴)) ≈ 𝐴)
5	2, 4	jca 554	. . . 4 ⊢ (𝐴 ∈ Fin → ((#‘𝐴) ∈ ℕ0 ∧ (1...(#‘𝐴)) ≈ 𝐴))
6		eleq1 2689	. . . . . 6 ⊢ (𝑛 = (#‘𝐴) → (𝑛 ∈ ℕ0 ↔ (#‘𝐴) ∈ ℕ0))
7		oveq2 6658	. . . . . . 7 ⊢ (𝑛 = (#‘𝐴) → (1...𝑛) = (1...(#‘𝐴)))
8	7	breq1d 4663	. . . . . 6 ⊢ (𝑛 = (#‘𝐴) → ((1...𝑛) ≈ 𝐴 ↔ (1...(#‘𝐴)) ≈ 𝐴))
9	6, 8	anbi12d 747	. . . . 5 ⊢ (𝑛 = (#‘𝐴) → ((𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴) ↔ ((#‘𝐴) ∈ ℕ0 ∧ (1...(#‘𝐴)) ≈ 𝐴)))
10	9	spcegv 3294	. . . 4 ⊢ ((#‘𝐴) ∈ V → (((#‘𝐴) ∈ ℕ0 ∧ (1...(#‘𝐴)) ≈ 𝐴) → ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)))
11	1, 5, 10	mpsyl 68	. . 3 ⊢ (𝐴 ∈ Fin → ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴))
12		df-rex 2918	. . 3 ⊢ (∃𝑛 ∈ ℕ0 (1...𝑛) ≈ 𝐴 ↔ ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴))
13	11, 12	sylibr 224	. 2 ⊢ (𝐴 ∈ Fin → ∃𝑛 ∈ ℕ0 (1...𝑛) ≈ 𝐴)
14		hasheni 13136	. . . . . . 7 ⊢ ((1...𝑛) ≈ 𝐴 → (#‘(1...𝑛)) = (#‘𝐴))
15	14	eqcomd 2628	. . . . . 6 ⊢ ((1...𝑛) ≈ 𝐴 → (#‘𝐴) = (#‘(1...𝑛)))
16		hashfz1 13134	. . . . . 6 ⊢ (𝑛 ∈ ℕ0 → (#‘(1...𝑛)) = 𝑛)
17		ovex 6678	. . . . . . 7 ⊢ (1...(#‘𝐴)) ∈ V
18		eqtr 2641	. . . . . . 7 ⊢ (((#‘𝐴) = (#‘(1...𝑛)) ∧ (#‘(1...𝑛)) = 𝑛) → (#‘𝐴) = 𝑛)
19		oveq2 6658	. . . . . . . 8 ⊢ ((#‘𝐴) = 𝑛 → (1...(#‘𝐴)) = (1...𝑛))
20		eqeng 7989	. . . . . . . 8 ⊢ ((1...(#‘𝐴)) ∈ V → ((1...(#‘𝐴)) = (1...𝑛) → (1...(#‘𝐴)) ≈ (1...𝑛)))
21	19, 20	syl5 34	. . . . . . 7 ⊢ ((1...(#‘𝐴)) ∈ V → ((#‘𝐴) = 𝑛 → (1...(#‘𝐴)) ≈ (1...𝑛)))
22	17, 18, 21	mpsyl 68	. . . . . 6 ⊢ (((#‘𝐴) = (#‘(1...𝑛)) ∧ (#‘(1...𝑛)) = 𝑛) → (1...(#‘𝐴)) ≈ (1...𝑛))
23	15, 16, 22	syl2anr 495	. . . . 5 ⊢ ((𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴) → (1...(#‘𝐴)) ≈ (1...𝑛))
24		entr 8008	. . . . 5 ⊢ (((1...(#‘𝐴)) ≈ (1...𝑛) ∧ (1...𝑛) ≈ 𝐴) → (1...(#‘𝐴)) ≈ 𝐴)
25	23, 24	sylancom 701	. . . 4 ⊢ ((𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴) → (1...(#‘𝐴)) ≈ 𝐴)
26	25, 3	sylibr 224	. . 3 ⊢ ((𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴) → 𝐴 ∈ Fin)
27	26	rexlimiva 3028	. 2 ⊢ (∃𝑛 ∈ ℕ0 (1...𝑛) ≈ 𝐴 → 𝐴 ∈ Fin)
28	13, 27	impbii 199	1 ⊢ (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ℕ0 (1...𝑛) ≈ 𝐴)

Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∃wrex 2913  Vcvv 3200   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650   ≈ cen 7952  Fincfn 7955  1c1 9937  ℕ₀cn0 11292  ...cfz 12326  #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-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-hash 13118

This theorem is referenced by:  rp-isfinite6  37864