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Theorem ctssdclemr 7105
Description: Lemma for ctssdc 7106. 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 5430 . . 3 (𝑓 = 𝑔 → (𝑓:ω–onto→(𝐴 ⊔ 1o) ↔ 𝑔:ω–onto→(𝐴 ⊔ 1o)))
21cbvexv 1918 . 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 2177 . . . . . 6 {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} = {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}
5 eqid 2177 . . . . . 6 (inl ∘ 𝑔) = (inl ∘ 𝑔)
63, 4, 5ctssdccl 7104 . . . . 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 7051 . . . . . . . . 9 inl:V–1-1-onto→({∅} × V)
8 f1ocnv 5470 . . . . . . . . 9 (inl:V–1-1-onto→({∅} × V) → inl:({∅} × V)–1-1-onto→V)
9 f1ofun 5459 . . . . . . . . 9 (inl:({∅} × V)–1-1-onto→V → Fun inl)
107, 8, 9mp2b 8 . . . . . . . 8 Fun inl
11 vex 2740 . . . . . . . 8 𝑔 ∈ V
12 cofunexg 6104 . . . . . . . 8 ((Fun inl ∧ 𝑔 ∈ V) → (inl ∘ 𝑔) ∈ V)
1310, 11, 12mp2an 426 . . . . . . 7 (inl ∘ 𝑔) ∈ V
14 foeq1 5430 . . . . . . 7 (𝑓 = (inl ∘ 𝑔) → (𝑓:{𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}–onto𝐴 ↔ (inl ∘ 𝑔):{𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}–onto𝐴))
1513, 14spcev 2832 . . . . . 6 ((inl ∘ 𝑔):{𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}–onto𝐴 → ∃𝑓 𝑓:{𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}–onto𝐴)
16153anim2i 1186 . . . . 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 4589 . . . . . 6 ω ∈ V
1918rabex 4144 . . . . 5 {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} ∈ V
20 sseq1 3178 . . . . . 6 (𝑠 = {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} → (𝑠 ⊆ ω ↔ {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} ⊆ ω))
21 foeq2 5431 . . . . . . 7 (𝑠 = {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} → (𝑓:𝑠onto𝐴𝑓:{𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}–onto𝐴))
2221exbidv 1825 . . . . . 6 (𝑠 = {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} → (∃𝑓 𝑓:𝑠onto𝐴 ↔ ∃𝑓 𝑓:{𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}–onto𝐴))
23 eleq2 2241 . . . . . . . 8 (𝑠 = {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} → (𝑛𝑠𝑛 ∈ {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}))
2423dcbid 838 . . . . . . 7 (𝑠 = {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} → (DECID 𝑛𝑠DECID 𝑛 ∈ {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}))
2524ralbidv 2477 . . . . . 6 (𝑠 = {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} → (∀𝑛 ∈ ω DECID 𝑛𝑠 ↔ ∀𝑛 ∈ ω DECID 𝑛 ∈ {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}))
2620, 22, 253anbi123d 1312 . . . . 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 2832 . . . 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 1598 . 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 834  w3a 978   = wceq 1353  wex 1492  wcel 2148  wral 2455  {crab 2459  Vcvv 2737  wss 3129  c0 3422  {csn 3591  ωcom 4586   × cxp 4621  ccnv 4622  cima 4626  ccom 4627  Fun wfun 5206  ontowfo 5210  1-1-ontowf1o 5211  cfv 5212  1oc1o 6404  cdju 7030  inlcinl 7038
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-1st 6135  df-2nd 6136  df-1o 6411  df-dju 7031  df-inl 7040  df-inr 7041
This theorem is referenced by:  ctssdc  7106
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