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Theorem enumct 7060
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 519 . . . . . . . . 9 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ 𝑛 = ∅) → 𝑓:𝑛onto𝐴)
2 foeq2 5390 . . . . . . . . . 10 (𝑛 = ∅ → (𝑓:𝑛onto𝐴𝑓:∅–onto𝐴))
32adantl 275 . . . . . . . . 9 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ 𝑛 = ∅) → (𝑓:𝑛onto𝐴𝑓:∅–onto𝐴))
41, 3mpbid 146 . . . . . . . 8 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ 𝑛 = ∅) → 𝑓:∅–onto𝐴)
5 fo00 5451 . . . . . . . 8 (𝑓:∅–onto𝐴 ↔ (𝑓 = ∅ ∧ 𝐴 = ∅))
64, 5sylib 121 . . . . . . 7 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ 𝑛 = ∅) → (𝑓 = ∅ ∧ 𝐴 = ∅))
7 0ct 7052 . . . . . . . 8 𝑔 𝑔:ω–onto→(∅ ⊔ 1o)
8 djueq1 6985 . . . . . . . . . 10 (𝐴 = ∅ → (𝐴 ⊔ 1o) = (∅ ⊔ 1o))
9 foeq3 5391 . . . . . . . . . 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 1805 . . . . . . . 8 (𝐴 = ∅ → (∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o) ↔ ∃𝑔 𝑔:ω–onto→(∅ ⊔ 1o)))
127, 11mpbiri 167 . . . . . . 7 (𝐴 = ∅ → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
136, 12simpl2im 384 . . . . . 6 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ 𝑛 = ∅) → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
14 omex 4553 . . . . . . . . 9 ω ∈ V
1514mptex 5694 . . . . . . . 8 (𝑘 ∈ ω ↦ if(𝑘𝑛, (𝑓𝑘), (𝑓‘∅))) ∈ V
16 simpll 519 . . . . . . . . 9 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → 𝑓:𝑛onto𝐴)
17 simplr 520 . . . . . . . . 9 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → 𝑛 ∈ ω)
18 simpr 109 . . . . . . . . 9 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → ∅ ∈ 𝑛)
19 eqid 2157 . . . . . . . . 9 (𝑘 ∈ ω ↦ if(𝑘𝑛, (𝑓𝑘), (𝑓‘∅))) = (𝑘 ∈ ω ↦ if(𝑘𝑛, (𝑓𝑘), (𝑓‘∅)))
2016, 17, 18, 19enumctlemm 7059 . . . . . . . 8 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → (𝑘 ∈ ω ↦ if(𝑘𝑛, (𝑓𝑘), (𝑓‘∅))):ω–onto𝐴)
21 foeq1 5389 . . . . . . . . 9 (𝑔 = (𝑘 ∈ ω ↦ if(𝑘𝑛, (𝑓𝑘), (𝑓‘∅))) → (𝑔:ω–onto𝐴 ↔ (𝑘 ∈ ω ↦ if(𝑘𝑛, (𝑓𝑘), (𝑓‘∅))):ω–onto𝐴))
2221spcegv 2800 . . . . . . . 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 5393 . . . . . . . . . . 11 (𝑓:𝑛onto𝐴𝑓:𝑛𝐴)
2524ad2antrr 480 . . . . . . . . . 10 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → 𝑓:𝑛𝐴)
2625, 18ffvelrnd 5604 . . . . . . . . 9 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → (𝑓‘∅) ∈ 𝐴)
27 eleq1 2220 . . . . . . . . . 10 (𝑥 = (𝑓‘∅) → (𝑥𝐴 ↔ (𝑓‘∅) ∈ 𝐴))
2827spcegv 2800 . . . . . . . . 9 ((𝑓‘∅) ∈ 𝐴 → ((𝑓‘∅) ∈ 𝐴 → ∃𝑥 𝑥𝐴))
2926, 26, 28sylc 62 . . . . . . . 8 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → ∃𝑥 𝑥𝐴)
30 ctm 7054 . . . . . . . 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 166 . . . . . 6 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
33 0elnn 4579 . . . . . . 7 (𝑛 ∈ ω → (𝑛 = ∅ ∨ ∅ ∈ 𝑛))
3433adantl 275 . . . . . 6 ((𝑓:𝑛onto𝐴𝑛 ∈ ω) → (𝑛 = ∅ ∨ ∅ ∈ 𝑛))
3513, 32, 34mpjaodan 788 . . . . 5 ((𝑓:𝑛onto𝐴𝑛 ∈ ω) → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
3635ex 114 . . . 4 (𝑓:𝑛onto𝐴 → (𝑛 ∈ ω → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o)))
3736exlimiv 1578 . . 3 (∃𝑓 𝑓:𝑛onto𝐴 → (𝑛 ∈ ω → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o)))
3837impcom 124 . 2 ((𝑛 ∈ ω ∧ ∃𝑓 𝑓:𝑛onto𝐴) → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
3938rexlimiva 2569 1 (∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛onto𝐴 → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wo 698   = wceq 1335  wex 1472  wcel 2128  wrex 2436  Vcvv 2712  c0 3394  ifcif 3505  cmpt 4026  ωcom 4550  wf 5167  ontowfo 5169  cfv 5171  1oc1o 6357  cdju 6982
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4080  ax-sep 4083  ax-nul 4091  ax-pow 4136  ax-pr 4170  ax-un 4394  ax-setind 4497  ax-iinf 4548
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-int 3809  df-iun 3852  df-br 3967  df-opab 4027  df-mpt 4028  df-tr 4064  df-id 4254  df-iord 4327  df-on 4329  df-suc 4332  df-iom 4551  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600  df-iota 5136  df-fun 5173  df-fn 5174  df-f 5175  df-f1 5176  df-fo 5177  df-f1o 5178  df-fv 5179  df-1st 6089  df-2nd 6090  df-1o 6364  df-dju 6983  df-inl 6992  df-inr 6993  df-case 7029
This theorem is referenced by:  finct  7061
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