Theorem enumct 7080

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 519	. . . . . . . . 9 ⊢ (((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) ∧ 𝑛 = ∅) → 𝑓:𝑛–onto→𝐴)
2		foeq2 5407	. . . . . . . . . 10 ⊢ (𝑛 = ∅ → (𝑓:𝑛–onto→𝐴 ↔ 𝑓:∅–onto→𝐴))
3	2	adantl 275	. . . . . . . . 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 5468	. . . . . . . 8 ⊢ (𝑓:∅–onto→𝐴 ↔ (𝑓 = ∅ ∧ 𝐴 = ∅))
6	4, 5	sylib 121	. . . . . . 7 ⊢ (((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) ∧ 𝑛 = ∅) → (𝑓 = ∅ ∧ 𝐴 = ∅))
7		0ct 7072	. . . . . . . 8 ⊢ ∃𝑔 𝑔:ω–onto→(∅ ⊔ 1o)
8		djueq1 7005	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → (𝐴 ⊔ 1o) = (∅ ⊔ 1o))
9		foeq3 5408	. . . . . . . . . 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 1813	. . . . . . . 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 384	. . . . . 6 ⊢ (((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) ∧ 𝑛 = ∅) → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
14		omex 4570	. . . . . . . . 9 ⊢ ω ∈ V
15	14	mptex 5711	. . . . . . . 8 ⊢ (𝑘 ∈ ω ↦ if(𝑘 ∈ 𝑛, (𝑓‘𝑘), (𝑓‘∅))) ∈ V
16		simpll 519	. . . . . . . . 9 ⊢ (((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → 𝑓:𝑛–onto→𝐴)
17		simplr 520	. . . . . . . . 9 ⊢ (((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → 𝑛 ∈ ω)
18		simpr 109	. . . . . . . . 9 ⊢ (((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → ∅ ∈ 𝑛)
19		eqid 2165	. . . . . . . . 9 ⊢ (𝑘 ∈ ω ↦ if(𝑘 ∈ 𝑛, (𝑓‘𝑘), (𝑓‘∅))) = (𝑘 ∈ ω ↦ if(𝑘 ∈ 𝑛, (𝑓‘𝑘), (𝑓‘∅)))
20	16, 17, 18, 19	enumctlemm 7079	. . . . . . . 8 ⊢ (((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → (𝑘 ∈ ω ↦ if(𝑘 ∈ 𝑛, (𝑓‘𝑘), (𝑓‘∅))):ω–onto→𝐴)
21		foeq1 5406	. . . . . . . . 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 2814	. . . . . . . 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 5410	. . . . . . . . . . 11 ⊢ (𝑓:𝑛–onto→𝐴 → 𝑓:𝑛⟶𝐴)
25	24	ad2antrr 480	. . . . . . . . . 10 ⊢ (((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → 𝑓:𝑛⟶𝐴)
26	25, 18	ffvelrnd 5621	. . . . . . . . 9 ⊢ (((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → (𝑓‘∅) ∈ 𝐴)
27		eleq1 2229	. . . . . . . . . 10 ⊢ (𝑥 = (𝑓‘∅) → (𝑥 ∈ 𝐴 ↔ (𝑓‘∅) ∈ 𝐴))
28	27	spcegv 2814	. . . . . . . . 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 7074	. . . . . . . 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 4596	. . . . . . 7 ⊢ (𝑛 ∈ ω → (𝑛 = ∅ ∨ ∅ ∈ 𝑛))
34	33	adantl 275	. . . . . 6 ⊢ ((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) → (𝑛 = ∅ ∨ ∅ ∈ 𝑛))
35	13, 32, 34	mpjaodan 788	. . . . 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 1586	. . 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 2578	1 ⊢ (∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛–onto→𝐴 → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698   = wceq 1343  ∃wex 1480   ∈ wcel 2136  ∃wrex 2445  Vcvv 2726  ∅c0 3409  ifcif 3520   ↦ cmpt 4043  ωcom 4567  ⟶wf 5184  –onto→wfo 5186  ‘cfv 5188  1_oc1o 6377   ⊔ cdju 7002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-1st 6108  df-2nd 6109  df-1o 6384  df-dju 7003  df-inl 7012  df-inr 7013  df-case 7049

This theorem is referenced by:  finct  7081