Theorem enumct 7114

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 527	. . . . . . . . 9 ⊢ (((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) ∧ 𝑛 = ∅) → 𝑓:𝑛–onto→𝐴)
2		foeq2 5436	. . . . . . . . . 10 ⊢ (𝑛 = ∅ → (𝑓:𝑛–onto→𝐴 ↔ 𝑓:∅–onto→𝐴))
3	2	adantl 277	. . . . . . . . 9 ⊢ (((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) ∧ 𝑛 = ∅) → (𝑓:𝑛–onto→𝐴 ↔ 𝑓:∅–onto→𝐴))
4	1, 3	mpbid 147	. . . . . . . 8 ⊢ (((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) ∧ 𝑛 = ∅) → 𝑓:∅–onto→𝐴)
5		fo00 5498	. . . . . . . 8 ⊢ (𝑓:∅–onto→𝐴 ↔ (𝑓 = ∅ ∧ 𝐴 = ∅))
6	4, 5	sylib 122	. . . . . . 7 ⊢ (((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) ∧ 𝑛 = ∅) → (𝑓 = ∅ ∧ 𝐴 = ∅))
7		0ct 7106	. . . . . . . 8 ⊢ ∃𝑔 𝑔:ω–onto→(∅ ⊔ 1o)
8		djueq1 7039	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → (𝐴 ⊔ 1o) = (∅ ⊔ 1o))
9		foeq3 5437	. . . . . . . . . 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 1825	. . . . . . . 8 ⊢ (𝐴 = ∅ → (∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o) ↔ ∃𝑔 𝑔:ω–onto→(∅ ⊔ 1o)))
12	7, 11	mpbiri 168	. . . . . . 7 ⊢ (𝐴 = ∅ → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
13	6, 12	simpl2im 386	. . . . . 6 ⊢ (((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) ∧ 𝑛 = ∅) → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
14		omex 4593	. . . . . . . . 9 ⊢ ω ∈ V
15	14	mptex 5743	. . . . . . . 8 ⊢ (𝑘 ∈ ω ↦ if(𝑘 ∈ 𝑛, (𝑓‘𝑘), (𝑓‘∅))) ∈ V
16		simpll 527	. . . . . . . . 9 ⊢ (((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → 𝑓:𝑛–onto→𝐴)
17		simplr 528	. . . . . . . . 9 ⊢ (((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → 𝑛 ∈ ω)
18		simpr 110	. . . . . . . . 9 ⊢ (((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → ∅ ∈ 𝑛)
19		eqid 2177	. . . . . . . . 9 ⊢ (𝑘 ∈ ω ↦ if(𝑘 ∈ 𝑛, (𝑓‘𝑘), (𝑓‘∅))) = (𝑘 ∈ ω ↦ if(𝑘 ∈ 𝑛, (𝑓‘𝑘), (𝑓‘∅)))
20	16, 17, 18, 19	enumctlemm 7113	. . . . . . . 8 ⊢ (((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → (𝑘 ∈ ω ↦ if(𝑘 ∈ 𝑛, (𝑓‘𝑘), (𝑓‘∅))):ω–onto→𝐴)
21		foeq1 5435	. . . . . . . . 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 2826	. . . . . . . 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 5439	. . . . . . . . . . 11 ⊢ (𝑓:𝑛–onto→𝐴 → 𝑓:𝑛⟶𝐴)
25	24	ad2antrr 488	. . . . . . . . . 10 ⊢ (((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → 𝑓:𝑛⟶𝐴)
26	25, 18	ffvelcdmd 5653	. . . . . . . . 9 ⊢ (((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → (𝑓‘∅) ∈ 𝐴)
27		eleq1 2240	. . . . . . . . . 10 ⊢ (𝑥 = (𝑓‘∅) → (𝑥 ∈ 𝐴 ↔ (𝑓‘∅) ∈ 𝐴))
28	27	spcegv 2826	. . . . . . . . 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 7108	. . . . . . . 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 167	. . . . . 6 ⊢ (((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
33		0elnn 4619	. . . . . . 7 ⊢ (𝑛 ∈ ω → (𝑛 = ∅ ∨ ∅ ∈ 𝑛))
34	33	adantl 277	. . . . . 6 ⊢ ((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) → (𝑛 = ∅ ∨ ∅ ∈ 𝑛))
35	13, 32, 34	mpjaodan 798	. . . . 5 ⊢ ((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
36	35	ex 115	. . . 4 ⊢ (𝑓:𝑛–onto→𝐴 → (𝑛 ∈ ω → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o)))
37	36	exlimiv 1598	. . 3 ⊢ (∃𝑓 𝑓:𝑛–onto→𝐴 → (𝑛 ∈ ω → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o)))
38	37	impcom 125	. 2 ⊢ ((𝑛 ∈ ω ∧ ∃𝑓 𝑓:𝑛–onto→𝐴) → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
39	38	rexlimiva 2589	1 ⊢ (∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛–onto→𝐴 → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 708   = wceq 1353  ∃wex 1492   ∈ wcel 2148  ∃wrex 2456  Vcvv 2738  ∅c0 3423  ifcif 3535   ↦ cmpt 4065  ωcom 4590  ⟶wf 5213  –onto→wfo 5215  ‘cfv 5217  1_oc1o 6410   ⊔ cdju 7036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-1st 6141  df-2nd 6142  df-1o 6417  df-dju 7037  df-inl 7046  df-inr 7047  df-case 7083

This theorem is referenced by:  finct  7115