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

Theorem enumct 7127
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 5447 . . . . . . . . . 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 5509 . . . . . . . 8 (𝑓:∅–onto𝐴 ↔ (𝑓 = ∅ ∧ 𝐴 = ∅))
64, 5sylib 122 . . . . . . 7 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ 𝑛 = ∅) → (𝑓 = ∅ ∧ 𝐴 = ∅))
7 0ct 7119 . . . . . . . 8 𝑔 𝑔:ω–onto→(∅ ⊔ 1o)
8 djueq1 7052 . . . . . . . . . 10 (𝐴 = ∅ → (𝐴 ⊔ 1o) = (∅ ⊔ 1o))
9 foeq3 5448 . . . . . . . . . 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 1835 . . . . . . . 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 4604 . . . . . . . . 9 ω ∈ V
1514mptex 5755 . . . . . . . 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 2187 . . . . . . . . 9 (𝑘 ∈ ω ↦ if(𝑘𝑛, (𝑓𝑘), (𝑓‘∅))) = (𝑘 ∈ ω ↦ if(𝑘𝑛, (𝑓𝑘), (𝑓‘∅)))
2016, 17, 18, 19enumctlemm 7126 . . . . . . . 8 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → (𝑘 ∈ ω ↦ if(𝑘𝑛, (𝑓𝑘), (𝑓‘∅))):ω–onto𝐴)
21 foeq1 5446 . . . . . . . . 9 (𝑔 = (𝑘 ∈ ω ↦ if(𝑘𝑛, (𝑓𝑘), (𝑓‘∅))) → (𝑔:ω–onto𝐴 ↔ (𝑘 ∈ ω ↦ if(𝑘𝑛, (𝑓𝑘), (𝑓‘∅))):ω–onto𝐴))
2221spcegv 2837 . . . . . . . 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 5450 . . . . . . . . . . 11 (𝑓:𝑛onto𝐴𝑓:𝑛𝐴)
2524ad2antrr 488 . . . . . . . . . 10 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → 𝑓:𝑛𝐴)
2625, 18ffvelcdmd 5665 . . . . . . . . 9 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → (𝑓‘∅) ∈ 𝐴)
27 eleq1 2250 . . . . . . . . . 10 (𝑥 = (𝑓‘∅) → (𝑥𝐴 ↔ (𝑓‘∅) ∈ 𝐴))
2827spcegv 2837 . . . . . . . . 9 ((𝑓‘∅) ∈ 𝐴 → ((𝑓‘∅) ∈ 𝐴 → ∃𝑥 𝑥𝐴))
2926, 26, 28sylc 62 . . . . . . . 8 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → ∃𝑥 𝑥𝐴)
30 ctm 7121 . . . . . . . 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 4630 . . . . . . 7 (𝑛 ∈ ω → (𝑛 = ∅ ∨ ∅ ∈ 𝑛))
3433adantl 277 . . . . . 6 ((𝑓:𝑛onto𝐴𝑛 ∈ ω) → (𝑛 = ∅ ∨ ∅ ∈ 𝑛))
3513, 32, 34mpjaodan 799 . . . . 5 ((𝑓:𝑛onto𝐴𝑛 ∈ ω) → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
3635ex 115 . . . 4 (𝑓:𝑛onto𝐴 → (𝑛 ∈ ω → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o)))
3736exlimiv 1608 . . 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 2599 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 709   = wceq 1363  wex 1502  wcel 2158  wrex 2466  Vcvv 2749  c0 3434  ifcif 3546  cmpt 4076  ωcom 4601  wf 5224  ontowfo 5226  cfv 5228  1oc1o 6423  cdju 7049
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-1st 6154  df-2nd 6155  df-1o 6430  df-dju 7050  df-inl 7059  df-inr 7060  df-case 7096
This theorem is referenced by:  finct  7128
  Copyright terms: Public domain W3C validator