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Theorem ctssdclemr 7007
 Description: Lemma for ctssdc 7008. 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 5350 . . 3 (𝑓 = 𝑔 → (𝑓:ω–onto→(𝐴 ⊔ 1o) ↔ 𝑔:ω–onto→(𝐴 ⊔ 1o)))
21cbvexv 1891 . 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 2140 . . . . . 6 {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} = {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}
5 eqid 2140 . . . . . 6 (inl ∘ 𝑔) = (inl ∘ 𝑔)
63, 4, 5ctssdccl 7006 . . . . 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 6953 . . . . . . . . 9 inl:V–1-1-onto→({∅} × V)
8 f1ocnv 5389 . . . . . . . . 9 (inl:V–1-1-onto→({∅} × V) → inl:({∅} × V)–1-1-onto→V)
9 f1ofun 5378 . . . . . . . . 9 (inl:({∅} × V)–1-1-onto→V → Fun inl)
107, 8, 9mp2b 8 . . . . . . . 8 Fun inl
11 vex 2693 . . . . . . . 8 𝑔 ∈ V
12 cofunexg 6018 . . . . . . . 8 ((Fun inl ∧ 𝑔 ∈ V) → (inl ∘ 𝑔) ∈ V)
1310, 11, 12mp2an 423 . . . . . . 7 (inl ∘ 𝑔) ∈ V
14 foeq1 5350 . . . . . . 7 (𝑓 = (inl ∘ 𝑔) → (𝑓:{𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}–onto𝐴 ↔ (inl ∘ 𝑔):{𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}–onto𝐴))
1513, 14spcev 2785 . . . . . 6 ((inl ∘ 𝑔):{𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}–onto𝐴 → ∃𝑓 𝑓:{𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}–onto𝐴)
16153anim2i 1169 . . . . 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 4516 . . . . . 6 ω ∈ V
1918rabex 4081 . . . . 5 {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} ∈ V
20 sseq1 3126 . . . . . 6 (𝑠 = {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} → (𝑠 ⊆ ω ↔ {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} ⊆ ω))
21 foeq2 5351 . . . . . . 7 (𝑠 = {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} → (𝑓:𝑠onto𝐴𝑓:{𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}–onto𝐴))
2221exbidv 1798 . . . . . 6 (𝑠 = {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} → (∃𝑓 𝑓:𝑠onto𝐴 ↔ ∃𝑓 𝑓:{𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}–onto𝐴))
23 eleq2 2204 . . . . . . . 8 (𝑠 = {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} → (𝑛𝑠𝑛 ∈ {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}))
2423dcbid 824 . . . . . . 7 (𝑠 = {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} → (DECID 𝑛𝑠DECID 𝑛 ∈ {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}))
2524ralbidv 2439 . . . . . 6 (𝑠 = {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} → (∀𝑛 ∈ ω DECID 𝑛𝑠 ↔ ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}))
2620, 22, 253anbi123d 1291 . . . . 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 2785 . . . 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 1578 . 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 820   ∧ w3a 963   = wceq 1332  ∃wex 1469   ∈ wcel 1481  ∀wral 2417  {crab 2421  Vcvv 2690   ⊆ wss 3077  ∅c0 3369  {csn 3533  ωcom 4513   × cxp 4546  ◡ccnv 4547   “ cima 4551   ∘ ccom 4552  Fun wfun 5126  –onto→wfo 5130  –1-1-onto→wf1o 5131  ‘cfv 5132  1oc1o 6315   ⊔ cdju 6932  inlcinl 6940 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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4052  ax-sep 4055  ax-nul 4063  ax-pow 4107  ax-pr 4140  ax-un 4364  ax-iinf 4511 This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2692  df-sbc 2915  df-csb 3009  df-dif 3079  df-un 3081  df-in 3083  df-ss 3090  df-nul 3370  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-int 3781  df-iun 3824  df-br 3939  df-opab 3999  df-mpt 4000  df-tr 4036  df-id 4224  df-iord 4297  df-on 4299  df-suc 4302  df-iom 4514  df-xp 4554  df-rel 4555  df-cnv 4556  df-co 4557  df-dm 4558  df-rn 4559  df-res 4560  df-ima 4561  df-iota 5097  df-fun 5134  df-fn 5135  df-f 5136  df-f1 5137  df-fo 5138  df-f1o 5139  df-fv 5140  df-1st 6047  df-2nd 6048  df-1o 6322  df-dju 6933  df-inl 6942  df-inr 6943 This theorem is referenced by:  ctssdc  7008
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