Theorem enumct 6873

Description: A finitely enumerable set is countable. Lemma 8.1.14 of [AczelRathjen], p. 73 (except that our definition of countable does not require the set to be inhabited). "Finitely enumerable" is defined as ∃𝑛 ∈ ω∃𝑓𝑓:𝑛–onto→𝐴 per Definition 8.1.4 of [AczelRathjen], p. 71 and "countable" is defined as ∃𝑔𝑔:ω–onto→(𝐴 ⊔ 1_o) per [BauerSwan], p. 14:3. (Contributed by Jim Kingdon, 13-Mar-2023.)

Assertion

Ref	Expression
enumct	⊢ (∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛–onto→𝐴 → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))

Distinct variable group:   𝐴,𝑓,𝑔,𝑛

Proof of Theorem enumct

Dummy variables 𝑥 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 497	. . . . . . . . 9 ⊢ (((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) ∧ 𝑛 = ∅) → 𝑓:𝑛–onto→𝐴)
2		foeq2 5265	. . . . . . . . . 10 ⊢ (𝑛 = ∅ → (𝑓:𝑛–onto→𝐴 ↔ 𝑓:∅–onto→𝐴))
3	2	adantl 272	. . . . . . . . 9 ⊢ (((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) ∧ 𝑛 = ∅) → (𝑓:𝑛–onto→𝐴 ↔ 𝑓:∅–onto→𝐴))
4	1, 3	mpbid 146	. . . . . . . 8 ⊢ (((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) ∧ 𝑛 = ∅) → 𝑓:∅–onto→𝐴)
5		fo00 5324	. . . . . . . 8 ⊢ (𝑓:∅–onto→𝐴 ↔ (𝑓 = ∅ ∧ 𝐴 = ∅))
6	4, 5	sylib 121	. . . . . . 7 ⊢ (((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) ∧ 𝑛 = ∅) → (𝑓 = ∅ ∧ 𝐴 = ∅))
7		0ct 6869	. . . . . . . 8 ⊢ ∃𝑔 𝑔:ω–onto→(∅ ⊔ 1o)
8		djueq1 6813	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → (𝐴 ⊔ 1o) = (∅ ⊔ 1o))
9		foeq3 5266	. . . . . . . . . 10 ⊢ ((𝐴 ⊔ 1o) = (∅ ⊔ 1o) → (𝑔:ω–onto→(𝐴 ⊔ 1o) ↔ 𝑔:ω–onto→(∅ ⊔ 1o)))
10	8, 9	syl 14	. . . . . . . . 9 ⊢ (𝐴 = ∅ → (𝑔:ω–onto→(𝐴 ⊔ 1o) ↔ 𝑔:ω–onto→(∅ ⊔ 1o)))
11	10	exbidv 1760	. . . . . . . 8 ⊢ (𝐴 = ∅ → (∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o) ↔ ∃𝑔 𝑔:ω–onto→(∅ ⊔ 1o)))
12	7, 11	mpbiri 167	. . . . . . 7 ⊢ (𝐴 = ∅ → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
13	6, 12	simpl2im 379	. . . . . 6 ⊢ (((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) ∧ 𝑛 = ∅) → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
14		omex 4436	. . . . . . . . 9 ⊢ ω ∈ V
15	14	mptex 5562	. . . . . . . 8 ⊢ (𝑘 ∈ ω ↦ if(𝑘 ∈ 𝑛, (𝑓‘𝑘), (𝑓‘∅))) ∈ V
16		simpll 497	. . . . . . . . 9 ⊢ (((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → 𝑓:𝑛–onto→𝐴)
17		simplr 498	. . . . . . . . 9 ⊢ (((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → 𝑛 ∈ ω)
18		simpr 109	. . . . . . . . 9 ⊢ (((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → ∅ ∈ 𝑛)
19		eqid 2095	. . . . . . . . 9 ⊢ (𝑘 ∈ ω ↦ if(𝑘 ∈ 𝑛, (𝑓‘𝑘), (𝑓‘∅))) = (𝑘 ∈ ω ↦ if(𝑘 ∈ 𝑛, (𝑓‘𝑘), (𝑓‘∅)))
20	16, 17, 18, 19	enumctlemm 6872	. . . . . . . 8 ⊢ (((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → (𝑘 ∈ ω ↦ if(𝑘 ∈ 𝑛, (𝑓‘𝑘), (𝑓‘∅))):ω–onto→𝐴)
21		foeq1 5264	. . . . . . . . 9 ⊢ (𝑔 = (𝑘 ∈ ω ↦ if(𝑘 ∈ 𝑛, (𝑓‘𝑘), (𝑓‘∅))) → (𝑔:ω–onto→𝐴 ↔ (𝑘 ∈ ω ↦ if(𝑘 ∈ 𝑛, (𝑓‘𝑘), (𝑓‘∅))):ω–onto→𝐴))
22	21	spcegv 2721	. . . . . . . 8 ⊢ ((𝑘 ∈ ω ↦ if(𝑘 ∈ 𝑛, (𝑓‘𝑘), (𝑓‘∅))) ∈ V → ((𝑘 ∈ ω ↦ if(𝑘 ∈ 𝑛, (𝑓‘𝑘), (𝑓‘∅))):ω–onto→𝐴 → ∃𝑔 𝑔:ω–onto→𝐴))
23	15, 20, 22	mpsyl 65	. . . . . . 7 ⊢ (((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → ∃𝑔 𝑔:ω–onto→𝐴)
24		fof 5268	. . . . . . . . . . 11 ⊢ (𝑓:𝑛–onto→𝐴 → 𝑓:𝑛⟶𝐴)
25	24	ad2antrr 473	. . . . . . . . . 10 ⊢ (((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → 𝑓:𝑛⟶𝐴)
26	25, 18	ffvelrnd 5474	. . . . . . . . 9 ⊢ (((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → (𝑓‘∅) ∈ 𝐴)
27		eleq1 2157	. . . . . . . . . 10 ⊢ (𝑥 = (𝑓‘∅) → (𝑥 ∈ 𝐴 ↔ (𝑓‘∅) ∈ 𝐴))
28	27	spcegv 2721	. . . . . . . . 9 ⊢ ((𝑓‘∅) ∈ 𝐴 → ((𝑓‘∅) ∈ 𝐴 → ∃𝑥 𝑥 ∈ 𝐴))
29	26, 26, 28	sylc 62	. . . . . . . 8 ⊢ (((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → ∃𝑥 𝑥 ∈ 𝐴)
30		ctm 6871	. . . . . . . 8 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o) ↔ ∃𝑔 𝑔:ω–onto→𝐴))
31	29, 30	syl 14	. . . . . . 7 ⊢ (((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → (∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o) ↔ ∃𝑔 𝑔:ω–onto→𝐴))
32	23, 31	mpbird 166	. . . . . 6 ⊢ (((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
33		0elnn 4460	. . . . . . 7 ⊢ (𝑛 ∈ ω → (𝑛 = ∅ ∨ ∅ ∈ 𝑛))
34	33	adantl 272	. . . . . 6 ⊢ ((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) → (𝑛 = ∅ ∨ ∅ ∈ 𝑛))
35	13, 32, 34	mpjaodan 750	. . . . 5 ⊢ ((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
36	35	ex 114	. . . 4 ⊢ (𝑓:𝑛–onto→𝐴 → (𝑛 ∈ ω → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o)))
37	36	exlimiv 1541	. . 3 ⊢ (∃𝑓 𝑓:𝑛–onto→𝐴 → (𝑛 ∈ ω → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o)))
38	37	impcom 124	. 2 ⊢ ((𝑛 ∈ ω ∧ ∃𝑓 𝑓:𝑛–onto→𝐴) → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
39	38	rexlimiva 2497	1 ⊢ (∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛–onto→𝐴 → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 667   = wceq 1296  ∃wex 1433   ∈ wcel 1445  ∃wrex 2371  Vcvv 2633  ∅c0 3302  ifcif 3413   ↦ cmpt 3921  ωcom 4433  ⟶wf 5045  –onto→wfo 5047  ‘cfv 5049  1_oc1o 6212   ⊔ cdju 6810

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431

This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-if 3414  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-id 4144  df-iord 4217  df-on 4219  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-1st 5949  df-2nd 5950  df-1o 6219  df-dju 6811  df-inl 6819  df-inr 6820  df-case 6855

This theorem is referenced by:  finct  6874