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Theorem ctssdclemr 6949
Description: Lemma for ctssdc 6950. 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 5299 . . 3 (𝑓 = 𝑔 → (𝑓:ω–onto→(𝐴 ⊔ 1o) ↔ 𝑔:ω–onto→(𝐴 ⊔ 1o)))
21cbvexv 1870 . 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 2115 . . . . . 6 {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} = {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}
5 eqid 2115 . . . . . 6 (inl ∘ 𝑔) = (inl ∘ 𝑔)
63, 4, 5ctssdccl 6948 . . . . 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 6895 . . . . . . . . 9 inl:V–1-1-onto→({∅} × V)
8 f1ocnv 5336 . . . . . . . . 9 (inl:V–1-1-onto→({∅} × V) → inl:({∅} × V)–1-1-onto→V)
9 f1ofun 5325 . . . . . . . . 9 (inl:({∅} × V)–1-1-onto→V → Fun inl)
107, 8, 9mp2b 8 . . . . . . . 8 Fun inl
11 vex 2660 . . . . . . . 8 𝑔 ∈ V
12 cofunexg 5963 . . . . . . . 8 ((Fun inl ∧ 𝑔 ∈ V) → (inl ∘ 𝑔) ∈ V)
1310, 11, 12mp2an 420 . . . . . . 7 (inl ∘ 𝑔) ∈ V
14 foeq1 5299 . . . . . . 7 (𝑓 = (inl ∘ 𝑔) → (𝑓:{𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}–onto𝐴 ↔ (inl ∘ 𝑔):{𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}–onto𝐴))
1513, 14spcev 2751 . . . . . 6 ((inl ∘ 𝑔):{𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}–onto𝐴 → ∃𝑓 𝑓:{𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}–onto𝐴)
16153anim2i 1151 . . . . 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 4467 . . . . . 6 ω ∈ V
1918rabex 4032 . . . . 5 {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} ∈ V
20 sseq1 3086 . . . . . 6 (𝑠 = {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} → (𝑠 ⊆ ω ↔ {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} ⊆ ω))
21 foeq2 5300 . . . . . . 7 (𝑠 = {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} → (𝑓:𝑠onto𝐴𝑓:{𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}–onto𝐴))
2221exbidv 1779 . . . . . 6 (𝑠 = {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} → (∃𝑓 𝑓:𝑠onto𝐴 ↔ ∃𝑓 𝑓:{𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}–onto𝐴))
23 eleq2 2178 . . . . . . . 8 (𝑠 = {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} → (𝑛𝑠𝑛 ∈ {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}))
2423dcbid 806 . . . . . . 7 (𝑠 = {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} → (DECID 𝑛𝑠DECID 𝑛 ∈ {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}))
2524ralbidv 2411 . . . . . 6 (𝑠 = {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} → (∀𝑛 ∈ ω DECID 𝑛𝑠 ↔ ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}))
2620, 22, 253anbi123d 1273 . . . . 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 2751 . . . 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 1560 . 2 (∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o) → ∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠onto𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛𝑠))
302, 29sylbi 120 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 802  w3a 945   = wceq 1314  wex 1451  wcel 1463  wral 2390  {crab 2394  Vcvv 2657  wss 3037  c0 3329  {csn 3493  ωcom 4464   × cxp 4497  ccnv 4498  cima 4502  ccom 4503  Fun wfun 5075  ontowfo 5079  1-1-ontowf1o 5080  cfv 5081  1oc1o 6260  cdju 6874  inlcinl 6882
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-iinf 4462
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-id 4175  df-iord 4248  df-on 4250  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-1st 5992  df-2nd 5993  df-1o 6267  df-dju 6875  df-inl 6884  df-inr 6885
This theorem is referenced by:  ctssdc  6950
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