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Theorem ctssdclemr 7101
Description: Lemma for ctssdc 7102. 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 5426 . . 3 (𝑓 = 𝑔 → (𝑓:ω–onto→(𝐴 ⊔ 1o) ↔ 𝑔:ω–onto→(𝐴 ⊔ 1o)))
21cbvexv 1916 . 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 2175 . . . . . 6 {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} = {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}
5 eqid 2175 . . . . . 6 (inl ∘ 𝑔) = (inl ∘ 𝑔)
63, 4, 5ctssdccl 7100 . . . . 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 7047 . . . . . . . . 9 inl:V–1-1-onto→({∅} × V)
8 f1ocnv 5466 . . . . . . . . 9 (inl:V–1-1-onto→({∅} × V) → inl:({∅} × V)–1-1-onto→V)
9 f1ofun 5455 . . . . . . . . 9 (inl:({∅} × V)–1-1-onto→V → Fun inl)
107, 8, 9mp2b 8 . . . . . . . 8 Fun inl
11 vex 2738 . . . . . . . 8 𝑔 ∈ V
12 cofunexg 6100 . . . . . . . 8 ((Fun inl ∧ 𝑔 ∈ V) → (inl ∘ 𝑔) ∈ V)
1310, 11, 12mp2an 426 . . . . . . 7 (inl ∘ 𝑔) ∈ V
14 foeq1 5426 . . . . . . 7 (𝑓 = (inl ∘ 𝑔) → (𝑓:{𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}–onto𝐴 ↔ (inl ∘ 𝑔):{𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}–onto𝐴))
1513, 14spcev 2830 . . . . . 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 4586 . . . . . 6 ω ∈ V
1918rabex 4142 . . . . 5 {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} ∈ V
20 sseq1 3176 . . . . . 6 (𝑠 = {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} → (𝑠 ⊆ ω ↔ {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} ⊆ ω))
21 foeq2 5427 . . . . . . 7 (𝑠 = {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} → (𝑓:𝑠onto𝐴𝑓:{𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}–onto𝐴))
2221exbidv 1823 . . . . . 6 (𝑠 = {𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)} → (∃𝑓 𝑓:𝑠onto𝐴 ↔ ∃𝑓 𝑓:{𝑡 ∈ ω ∣ (𝑔𝑡) ∈ (inl “ 𝐴)}–onto𝐴))
23 eleq2 2239 . . . . . . . 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 2475 . . . . . 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 2830 . . . 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 1596 . 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 1490  wcel 2146  wral 2453  {crab 2457  Vcvv 2735  wss 3127  c0 3420  {csn 3589  ωcom 4583   × cxp 4618  ccnv 4619  cima 4623  ccom 4624  Fun wfun 5202  ontowfo 5206  1-1-ontowf1o 5207  cfv 5208  1oc1o 6400  cdju 7026  inlcinl 7034
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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-iinf 4581
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-iord 4360  df-on 4362  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-1st 6131  df-2nd 6132  df-1o 6407  df-dju 7027  df-inl 7036  df-inr 7037
This theorem is referenced by:  ctssdc  7102
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