ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  enumct GIF version

Theorem enumct 7111
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:𝑛onto𝐴 per Definition 8.1.4 of [AczelRathjen], p. 71 and "countable" is defined as 𝑔𝑔:ω–onto→(𝐴 ⊔ 1o) per [BauerSwan], p. 14:3. (Contributed by Jim Kingdon, 13-Mar-2023.)
Assertion
Ref Expression
enumct (∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛onto𝐴 → ∃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.
StepHypRef Expression
1 simpll 527 . . . . . . . . 9 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ 𝑛 = ∅) → 𝑓:𝑛onto𝐴)
2 foeq2 5434 . . . . . . . . . 10 (𝑛 = ∅ → (𝑓:𝑛onto𝐴𝑓:∅–onto𝐴))
32adantl 277 . . . . . . . . 9 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ 𝑛 = ∅) → (𝑓:𝑛onto𝐴𝑓:∅–onto𝐴))
41, 3mpbid 147 . . . . . . . 8 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ 𝑛 = ∅) → 𝑓:∅–onto𝐴)
5 fo00 5496 . . . . . . . 8 (𝑓:∅–onto𝐴 ↔ (𝑓 = ∅ ∧ 𝐴 = ∅))
64, 5sylib 122 . . . . . . 7 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ 𝑛 = ∅) → (𝑓 = ∅ ∧ 𝐴 = ∅))
7 0ct 7103 . . . . . . . 8 𝑔 𝑔:ω–onto→(∅ ⊔ 1o)
8 djueq1 7036 . . . . . . . . . 10 (𝐴 = ∅ → (𝐴 ⊔ 1o) = (∅ ⊔ 1o))
9 foeq3 5435 . . . . . . . . . 10 ((𝐴 ⊔ 1o) = (∅ ⊔ 1o) → (𝑔:ω–onto→(𝐴 ⊔ 1o) ↔ 𝑔:ω–onto→(∅ ⊔ 1o)))
108, 9syl 14 . . . . . . . . 9 (𝐴 = ∅ → (𝑔:ω–onto→(𝐴 ⊔ 1o) ↔ 𝑔:ω–onto→(∅ ⊔ 1o)))
1110exbidv 1825 . . . . . . . 8 (𝐴 = ∅ → (∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o) ↔ ∃𝑔 𝑔:ω–onto→(∅ ⊔ 1o)))
127, 11mpbiri 168 . . . . . . 7 (𝐴 = ∅ → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
136, 12simpl2im 386 . . . . . 6 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ 𝑛 = ∅) → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
14 omex 4591 . . . . . . . . 9 ω ∈ V
1514mptex 5741 . . . . . . . 8 (𝑘 ∈ ω ↦ if(𝑘𝑛, (𝑓𝑘), (𝑓‘∅))) ∈ V
16 simpll 527 . . . . . . . . 9 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → 𝑓:𝑛onto𝐴)
17 simplr 528 . . . . . . . . 9 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → 𝑛 ∈ ω)
18 simpr 110 . . . . . . . . 9 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → ∅ ∈ 𝑛)
19 eqid 2177 . . . . . . . . 9 (𝑘 ∈ ω ↦ if(𝑘𝑛, (𝑓𝑘), (𝑓‘∅))) = (𝑘 ∈ ω ↦ if(𝑘𝑛, (𝑓𝑘), (𝑓‘∅)))
2016, 17, 18, 19enumctlemm 7110 . . . . . . . 8 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → (𝑘 ∈ ω ↦ if(𝑘𝑛, (𝑓𝑘), (𝑓‘∅))):ω–onto𝐴)
21 foeq1 5433 . . . . . . . . 9 (𝑔 = (𝑘 ∈ ω ↦ if(𝑘𝑛, (𝑓𝑘), (𝑓‘∅))) → (𝑔:ω–onto𝐴 ↔ (𝑘 ∈ ω ↦ if(𝑘𝑛, (𝑓𝑘), (𝑓‘∅))):ω–onto𝐴))
2221spcegv 2825 . . . . . . . 8 ((𝑘 ∈ ω ↦ if(𝑘𝑛, (𝑓𝑘), (𝑓‘∅))) ∈ V → ((𝑘 ∈ ω ↦ if(𝑘𝑛, (𝑓𝑘), (𝑓‘∅))):ω–onto𝐴 → ∃𝑔 𝑔:ω–onto𝐴))
2315, 20, 22mpsyl 65 . . . . . . 7 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → ∃𝑔 𝑔:ω–onto𝐴)
24 fof 5437 . . . . . . . . . . 11 (𝑓:𝑛onto𝐴𝑓:𝑛𝐴)
2524ad2antrr 488 . . . . . . . . . 10 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → 𝑓:𝑛𝐴)
2625, 18ffvelcdmd 5651 . . . . . . . . 9 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → (𝑓‘∅) ∈ 𝐴)
27 eleq1 2240 . . . . . . . . . 10 (𝑥 = (𝑓‘∅) → (𝑥𝐴 ↔ (𝑓‘∅) ∈ 𝐴))
2827spcegv 2825 . . . . . . . . 9 ((𝑓‘∅) ∈ 𝐴 → ((𝑓‘∅) ∈ 𝐴 → ∃𝑥 𝑥𝐴))
2926, 26, 28sylc 62 . . . . . . . 8 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → ∃𝑥 𝑥𝐴)
30 ctm 7105 . . . . . . . 8 (∃𝑥 𝑥𝐴 → (∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o) ↔ ∃𝑔 𝑔:ω–onto𝐴))
3129, 30syl 14 . . . . . . 7 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → (∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o) ↔ ∃𝑔 𝑔:ω–onto𝐴))
3223, 31mpbird 167 . . . . . 6 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
33 0elnn 4617 . . . . . . 7 (𝑛 ∈ ω → (𝑛 = ∅ ∨ ∅ ∈ 𝑛))
3433adantl 277 . . . . . 6 ((𝑓:𝑛onto𝐴𝑛 ∈ ω) → (𝑛 = ∅ ∨ ∅ ∈ 𝑛))
3513, 32, 34mpjaodan 798 . . . . 5 ((𝑓:𝑛onto𝐴𝑛 ∈ ω) → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
3635ex 115 . . . 4 (𝑓:𝑛onto𝐴 → (𝑛 ∈ ω → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o)))
3736exlimiv 1598 . . 3 (∃𝑓 𝑓:𝑛onto𝐴 → (𝑛 ∈ ω → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o)))
3837impcom 125 . 2 ((𝑛 ∈ ω ∧ ∃𝑓 𝑓:𝑛onto𝐴) → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
3938rexlimiva 2589 1 (∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛onto𝐴 → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wo 708   = wceq 1353  wex 1492  wcel 2148  wrex 2456  Vcvv 2737  c0 3422  ifcif 3534  cmpt 4063  ωcom 4588  wf 5211  ontowfo 5213  cfv 5215  1oc1o 6407  cdju 7033
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 4117  ax-sep 4120  ax-nul 4128  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-iinf 4586
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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-tr 4101  df-id 4292  df-iord 4365  df-on 4367  df-suc 4370  df-iom 4589  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-1st 6138  df-2nd 6139  df-1o 6414  df-dju 7034  df-inl 7043  df-inr 7044  df-case 7080
This theorem is referenced by:  finct  7112
  Copyright terms: Public domain W3C validator