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Theorem enumct 7092
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 524 . . . . . . . . 9 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ 𝑛 = ∅) → 𝑓:𝑛onto𝐴)
2 foeq2 5417 . . . . . . . . . 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 5478 . . . . . . . 8 (𝑓:∅–onto𝐴 ↔ (𝑓 = ∅ ∧ 𝐴 = ∅))
64, 5sylib 121 . . . . . . 7 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ 𝑛 = ∅) → (𝑓 = ∅ ∧ 𝐴 = ∅))
7 0ct 7084 . . . . . . . 8 𝑔 𝑔:ω–onto→(∅ ⊔ 1o)
8 djueq1 7017 . . . . . . . . . 10 (𝐴 = ∅ → (𝐴 ⊔ 1o) = (∅ ⊔ 1o))
9 foeq3 5418 . . . . . . . . . 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 1818 . . . . . . . 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 4577 . . . . . . . . 9 ω ∈ V
1514mptex 5722 . . . . . . . 8 (𝑘 ∈ ω ↦ if(𝑘𝑛, (𝑓𝑘), (𝑓‘∅))) ∈ V
16 simpll 524 . . . . . . . . 9 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → 𝑓:𝑛onto𝐴)
17 simplr 525 . . . . . . . . 9 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → 𝑛 ∈ ω)
18 simpr 109 . . . . . . . . 9 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → ∅ ∈ 𝑛)
19 eqid 2170 . . . . . . . . 9 (𝑘 ∈ ω ↦ if(𝑘𝑛, (𝑓𝑘), (𝑓‘∅))) = (𝑘 ∈ ω ↦ if(𝑘𝑛, (𝑓𝑘), (𝑓‘∅)))
2016, 17, 18, 19enumctlemm 7091 . . . . . . . 8 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → (𝑘 ∈ ω ↦ if(𝑘𝑛, (𝑓𝑘), (𝑓‘∅))):ω–onto𝐴)
21 foeq1 5416 . . . . . . . . 9 (𝑔 = (𝑘 ∈ ω ↦ if(𝑘𝑛, (𝑓𝑘), (𝑓‘∅))) → (𝑔:ω–onto𝐴 ↔ (𝑘 ∈ ω ↦ if(𝑘𝑛, (𝑓𝑘), (𝑓‘∅))):ω–onto𝐴))
2221spcegv 2818 . . . . . . . 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 5420 . . . . . . . . . . 11 (𝑓:𝑛onto𝐴𝑓:𝑛𝐴)
2524ad2antrr 485 . . . . . . . . . 10 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → 𝑓:𝑛𝐴)
2625, 18ffvelrnd 5632 . . . . . . . . 9 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → (𝑓‘∅) ∈ 𝐴)
27 eleq1 2233 . . . . . . . . . 10 (𝑥 = (𝑓‘∅) → (𝑥𝐴 ↔ (𝑓‘∅) ∈ 𝐴))
2827spcegv 2818 . . . . . . . . 9 ((𝑓‘∅) ∈ 𝐴 → ((𝑓‘∅) ∈ 𝐴 → ∃𝑥 𝑥𝐴))
2926, 26, 28sylc 62 . . . . . . . 8 (((𝑓:𝑛onto𝐴𝑛 ∈ ω) ∧ ∅ ∈ 𝑛) → ∃𝑥 𝑥𝐴)
30 ctm 7086 . . . . . . . 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 4603 . . . . . . 7 (𝑛 ∈ ω → (𝑛 = ∅ ∨ ∅ ∈ 𝑛))
3433adantl 275 . . . . . 6 ((𝑓:𝑛onto𝐴𝑛 ∈ ω) → (𝑛 = ∅ ∨ ∅ ∈ 𝑛))
3513, 32, 34mpjaodan 793 . . . . 5 ((𝑓:𝑛onto𝐴𝑛 ∈ ω) → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
3635ex 114 . . . 4 (𝑓:𝑛onto𝐴 → (𝑛 ∈ ω → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o)))
3736exlimiv 1591 . . 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 2582 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 703   = wceq 1348  wex 1485  wcel 2141  wrex 2449  Vcvv 2730  c0 3414  ifcif 3526  cmpt 4050  ωcom 4574  wf 5194  ontowfo 5196  cfv 5198  1oc1o 6388  cdju 7014
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-1st 6119  df-2nd 6120  df-1o 6395  df-dju 7015  df-inl 7024  df-inr 7025  df-case 7061
This theorem is referenced by:  finct  7093
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