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Theorem ctssdclemr 7142
Description: Lemma for ctssdc 7143. Showing that our usual definition of countable implies the alternate one. (Contributed by Jim Kingdon, 16-Aug-2023.)
Assertion
Ref Expression
ctssdclemr (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) → ∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠onto𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛𝑠))
Distinct variable groups:   𝐴,𝑓,𝑠   𝐴,𝑛,𝑠

Proof of Theorem ctssdclemr
Dummy variables 𝑔 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 foeq1 5453 . . 3 (𝑓 = 𝑔 → (𝑓:ω–onto→(𝐴 ⊔ 1o) ↔ 𝑔:ω–onto→(𝐴 ⊔ 1o)))
21cbvexv 1930 . 2 (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) ↔ ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
3 id 19 . . . . . 6 (𝑔:ω–onto→(𝐴 ⊔ 1o) → 𝑔:ω–onto→(𝐴 ⊔ 1o))
4 eqid 2189 . . . . . 6 {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} = {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}
5 eqid 2189 . . . . . 6 (inl ∘ 𝑔) = (inl ∘ 𝑔)
63, 4, 5ctssdccl 7141 . . . . 5 (𝑔:ω–onto→(𝐴 ⊔ 1o) → ({𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} ⊆ ω ∧ (inl ∘ 𝑔):{𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}–onto𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}))
7 djulf1o 7088 . . . . . . . . 9 inl:V–1-1-onto→({∅} × V)
8 f1ocnv 5493 . . . . . . . . 9 (inl:V–1-1-onto→({∅} × V) → inl:({∅} × V)–1-1-onto→V)
9 f1ofun 5482 . . . . . . . . 9 (inl:({∅} × V)–1-1-onto→V → Fun inl)
107, 8, 9mp2b 8 . . . . . . . 8 Fun inl
11 vex 2755 . . . . . . . 8 𝑔 ∈ V
12 cofunexg 6135 . . . . . . . 8 ((Fun inl ∧ 𝑔 ∈ V) → (inl ∘ 𝑔) ∈ V)
1310, 11, 12mp2an 426 . . . . . . 7 (inl ∘ 𝑔) ∈ V
14 foeq1 5453 . . . . . . 7 (𝑓 = (inl ∘ 𝑔) → (𝑓:{𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}–onto𝐴 ↔ (inl ∘ 𝑔):{𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}–onto𝐴))
1513, 14spcev 2847 . . . . . 6 ((inl ∘ 𝑔):{𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}–onto𝐴 → ∃𝑓 𝑓:{𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}–onto𝐴)
16153anim2i 1188 . . . . 5 (({𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} ⊆ ω ∧ (inl ∘ 𝑔):{𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}–onto𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}) → ({𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} ⊆ ω ∧ ∃𝑓 𝑓:{𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}–onto𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}))
176, 16syl 14 . . . 4 (𝑔:ω–onto→(𝐴 ⊔ 1o) → ({𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} ⊆ ω ∧ ∃𝑓 𝑓:{𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}–onto𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}))
18 omex 4610 . . . . . 6 ω ∈ V
1918rabex 4162 . . . . 5 {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} ∈ V
20 sseq1 3193 . . . . . 6 (𝑠 = {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} → (𝑠 ⊆ ω ↔ {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} ⊆ ω))
21 foeq2 5454 . . . . . . 7 (𝑠 = {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} → (𝑓:𝑠onto𝐴𝑓:{𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}–onto𝐴))
2221exbidv 1836 . . . . . 6 (𝑠 = {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} → (∃𝑓 𝑓:𝑠onto𝐴 ↔ ∃𝑓 𝑓:{𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}–onto𝐴))
23 eleq2 2253 . . . . . . . 8 (𝑠 = {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} → (𝑛𝑠𝑛 ∈ {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}))
2423dcbid 839 . . . . . . 7 (𝑠 = {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} → (DECID 𝑛𝑠DECID 𝑛 ∈ {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}))
2524ralbidv 2490 . . . . . 6 (𝑠 = {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} → (∀𝑛 ∈ ω DECID 𝑛𝑠 ↔ ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}))
2620, 22, 253anbi123d 1323 . . . . 5 (𝑠 = {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} → ((𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠onto𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛𝑠) ↔ ({𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} ⊆ ω ∧ ∃𝑓 𝑓:{𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}–onto𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)})))
2719, 26spcev 2847 . . . 4 (({𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} ⊆ ω ∧ ∃𝑓 𝑓:{𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}–onto𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}) → ∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠onto𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛𝑠))
2817, 27syl 14 . . 3 (𝑔:ω–onto→(𝐴 ⊔ 1o) → ∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠onto𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛𝑠))
2928exlimiv 1609 . 2 (∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o) → ∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠onto𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛𝑠))
302, 29sylbi 121 1 (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) → ∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠onto𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛𝑠))
Colors of variables: wff set class
Syntax hints:  wi 4  DECID wdc 835  w3a 980   = wceq 1364  wex 1503  wcel 2160  wral 2468  {crab 2472  Vcvv 2752  wss 3144  c0 3437  {csn 3607  ωcom 4607   × cxp 4642  ccnv 4643  cima 4647  ccom 4648  Fun wfun 5229  ontowfo 5233  1-1-ontowf1o 5234  cfv 5235  1oc1o 6435  cdju 7067  inlcinl 7075
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-1st 6166  df-2nd 6167  df-1o 6442  df-dju 7068  df-inl 7077  df-inr 7078
This theorem is referenced by:  ctssdc  7143
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