Theorem enumct 7181

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 5477	. . . . . . . . . 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 5540	. . . . . . . 8 ⊢ (𝑓:∅–onto→𝐴 ↔ (𝑓 = ∅ ∧ 𝐴 = ∅))
6	4, 5	sylib 122	. . . . . . 7 ⊢ (((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) ∧ 𝑛 = ∅) → (𝑓 = ∅ ∧ 𝐴 = ∅))
7		0ct 7173	. . . . . . . 8 ⊢ ∃𝑔 𝑔:ω–onto→(∅ ⊔ 1o)
8		djueq1 7106	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → (𝐴 ⊔ 1o) = (∅ ⊔ 1o))
9		foeq3 5478	. . . . . . . . . 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 1839	. . . . . . . 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 4629	. . . . . . . . 9 ⊢ ω ∈ V
15	14	mptex 5788	. . . . . . . 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 2196	. . . . . . . . 9 ⊢ (𝑘 ∈ ω ↦ if(𝑘 ∈ 𝑛, (𝑓‘𝑘), (𝑓‘∅))) = (𝑘 ∈ ω ↦ if(𝑘 ∈ 𝑛, (𝑓‘𝑘), (𝑓‘∅)))
20	16, 17, 18, 19	enumctlemm 7180	. . . . . . . 8 ⊢ (((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → (𝑘 ∈ ω ↦ if(𝑘 ∈ 𝑛, (𝑓‘𝑘), (𝑓‘∅))):ω–onto→𝐴)
21		foeq1 5476	. . . . . . . . 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 2852	. . . . . . . 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 5480	. . . . . . . . . . 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 5698	. . . . . . . . 9 ⊢ (((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → (𝑓‘∅) ∈ 𝐴)
27		eleq1 2259	. . . . . . . . . 10 ⊢ (𝑥 = (𝑓‘∅) → (𝑥 ∈ 𝐴 ↔ (𝑓‘∅) ∈ 𝐴))
28	27	spcegv 2852	. . . . . . . . 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 7175	. . . . . . . 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 4655	. . . . . . 7 ⊢ (𝑛 ∈ ω → (𝑛 = ∅ ∨ ∅ ∈ 𝑛))
34	33	adantl 277	. . . . . 6 ⊢ ((𝑓:𝑛–onto→𝐴 ∧ 𝑛 ∈ ω) → (𝑛 = ∅ ∨ ∅ ∈ 𝑛))
35	13, 32, 34	mpjaodan 799	. . . . 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 1612	. . 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 2609	1 ⊢ (∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛–onto→𝐴 → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709   = wceq 1364  ∃wex 1506   ∈ wcel 2167  ∃wrex 2476  Vcvv 2763  ∅c0 3450  ifcif 3561   ↦ cmpt 4094  ωcom 4626  ⟶wf 5254  –onto→wfo 5256  ‘cfv 5258  1_oc1o 6467   ⊔ cdju 7103

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-1st 6198  df-2nd 6199  df-1o 6474  df-dju 7104  df-inl 7113  df-inr 7114  df-case 7150

This theorem is referenced by:  finct  7182