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Theorem ctm 7102
Description: Two equivalent definitions of countable for an inhabited set. Remark of [BauerSwan], p. 14:3. (Contributed by Jim Kingdon, 13-Mar-2023.)
Assertion
Ref Expression
ctm (∃𝑥 𝑥𝐴 → (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) ↔ ∃𝑓 𝑓:ω–onto𝐴))
Distinct variable group:   𝐴,𝑓,𝑥

Proof of Theorem ctm
Dummy variables 𝑔 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1oi 5495 . . . . . . . . . . 11 ( I ↾ 𝐴):𝐴1-1-onto𝐴
2 f1of 5457 . . . . . . . . . . 11 (( I ↾ 𝐴):𝐴1-1-onto𝐴 → ( I ↾ 𝐴):𝐴𝐴)
31, 2mp1i 10 . . . . . . . . . 10 ((𝑥𝐴𝑓:ω–onto→(𝐴 ⊔ 1o)) → ( I ↾ 𝐴):𝐴𝐴)
4 fconst6g 5410 . . . . . . . . . . 11 (𝑥𝐴 → (1o × {𝑥}):1o𝐴)
54adantr 276 . . . . . . . . . 10 ((𝑥𝐴𝑓:ω–onto→(𝐴 ⊔ 1o)) → (1o × {𝑥}):1o𝐴)
63, 5casef 7081 . . . . . . . . 9 ((𝑥𝐴𝑓:ω–onto→(𝐴 ⊔ 1o)) → case(( I ↾ 𝐴), (1o × {𝑥})):(𝐴 ⊔ 1o)⟶𝐴)
7 ffun 5364 . . . . . . . . 9 (case(( I ↾ 𝐴), (1o × {𝑥})):(𝐴 ⊔ 1o)⟶𝐴 → Fun case(( I ↾ 𝐴), (1o × {𝑥})))
86, 7syl 14 . . . . . . . 8 ((𝑥𝐴𝑓:ω–onto→(𝐴 ⊔ 1o)) → Fun case(( I ↾ 𝐴), (1o × {𝑥})))
9 vex 2740 . . . . . . . . 9 𝑓 ∈ V
109a1i 9 . . . . . . . 8 ((𝑥𝐴𝑓:ω–onto→(𝐴 ⊔ 1o)) → 𝑓 ∈ V)
11 cofunexg 6104 . . . . . . . 8 ((Fun case(( I ↾ 𝐴), (1o × {𝑥})) ∧ 𝑓 ∈ V) → (case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓) ∈ V)
128, 10, 11syl2anc 411 . . . . . . 7 ((𝑥𝐴𝑓:ω–onto→(𝐴 ⊔ 1o)) → (case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓) ∈ V)
13 fof 5434 . . . . . . . . . 10 (𝑓:ω–onto→(𝐴 ⊔ 1o) → 𝑓:ω⟶(𝐴 ⊔ 1o))
1413adantl 277 . . . . . . . . 9 ((𝑥𝐴𝑓:ω–onto→(𝐴 ⊔ 1o)) → 𝑓:ω⟶(𝐴 ⊔ 1o))
15 fco 5377 . . . . . . . . 9 ((case(( I ↾ 𝐴), (1o × {𝑥})):(𝐴 ⊔ 1o)⟶𝐴𝑓:ω⟶(𝐴 ⊔ 1o)) → (case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓):ω⟶𝐴)
166, 14, 15syl2anc 411 . . . . . . . 8 ((𝑥𝐴𝑓:ω–onto→(𝐴 ⊔ 1o)) → (case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓):ω⟶𝐴)
17 simplr 528 . . . . . . . . . . 11 (((𝑥𝐴𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦𝐴) → 𝑓:ω–onto→(𝐴 ⊔ 1o))
18 djulcl 7044 . . . . . . . . . . . 12 (𝑦𝐴 → (inl‘𝑦) ∈ (𝐴 ⊔ 1o))
1918adantl 277 . . . . . . . . . . 11 (((𝑥𝐴𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦𝐴) → (inl‘𝑦) ∈ (𝐴 ⊔ 1o))
20 foelrn 5748 . . . . . . . . . . 11 ((𝑓:ω–onto→(𝐴 ⊔ 1o) ∧ (inl‘𝑦) ∈ (𝐴 ⊔ 1o)) → ∃𝑧 ∈ ω (inl‘𝑦) = (𝑓𝑧))
2117, 19, 20syl2anc 411 . . . . . . . . . 10 (((𝑥𝐴𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦𝐴) → ∃𝑧 ∈ ω (inl‘𝑦) = (𝑓𝑧))
22 fofn 5436 . . . . . . . . . . . . . . . 16 (𝑓:ω–onto→(𝐴 ⊔ 1o) → 𝑓 Fn ω)
2322ad4antlr 495 . . . . . . . . . . . . . . 15 (((((𝑥𝐴𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦𝐴) ∧ 𝑧 ∈ ω) ∧ (inl‘𝑦) = (𝑓𝑧)) → 𝑓 Fn ω)
24 simplr 528 . . . . . . . . . . . . . . 15 (((((𝑥𝐴𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦𝐴) ∧ 𝑧 ∈ ω) ∧ (inl‘𝑦) = (𝑓𝑧)) → 𝑧 ∈ ω)
25 fvco2 5581 . . . . . . . . . . . . . . 15 ((𝑓 Fn ω ∧ 𝑧 ∈ ω) → ((case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓)‘𝑧) = (case(( I ↾ 𝐴), (1o × {𝑥}))‘(𝑓𝑧)))
2623, 24, 25syl2anc 411 . . . . . . . . . . . . . 14 (((((𝑥𝐴𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦𝐴) ∧ 𝑧 ∈ ω) ∧ (inl‘𝑦) = (𝑓𝑧)) → ((case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓)‘𝑧) = (case(( I ↾ 𝐴), (1o × {𝑥}))‘(𝑓𝑧)))
27 simpr 110 . . . . . . . . . . . . . . 15 (((((𝑥𝐴𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦𝐴) ∧ 𝑧 ∈ ω) ∧ (inl‘𝑦) = (𝑓𝑧)) → (inl‘𝑦) = (𝑓𝑧))
2827fveq2d 5515 . . . . . . . . . . . . . 14 (((((𝑥𝐴𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦𝐴) ∧ 𝑧 ∈ ω) ∧ (inl‘𝑦) = (𝑓𝑧)) → (case(( I ↾ 𝐴), (1o × {𝑥}))‘(inl‘𝑦)) = (case(( I ↾ 𝐴), (1o × {𝑥}))‘(𝑓𝑧)))
29 fnresi 5329 . . . . . . . . . . . . . . . 16 ( I ↾ 𝐴) Fn 𝐴
3029a1i 9 . . . . . . . . . . . . . . 15 (((((𝑥𝐴𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦𝐴) ∧ 𝑧 ∈ ω) ∧ (inl‘𝑦) = (𝑓𝑧)) → ( I ↾ 𝐴) Fn 𝐴)
31 vex 2740 . . . . . . . . . . . . . . . . 17 𝑥 ∈ V
3231fconst6 5411 . . . . . . . . . . . . . . . 16 (1o × {𝑥}):1o⟶V
33 ffun 5364 . . . . . . . . . . . . . . . 16 ((1o × {𝑥}):1o⟶V → Fun (1o × {𝑥}))
3432, 33mp1i 10 . . . . . . . . . . . . . . 15 (((((𝑥𝐴𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦𝐴) ∧ 𝑧 ∈ ω) ∧ (inl‘𝑦) = (𝑓𝑧)) → Fun (1o × {𝑥}))
35 simpllr 534 . . . . . . . . . . . . . . 15 (((((𝑥𝐴𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦𝐴) ∧ 𝑧 ∈ ω) ∧ (inl‘𝑦) = (𝑓𝑧)) → 𝑦𝐴)
3630, 34, 35caseinl 7084 . . . . . . . . . . . . . 14 (((((𝑥𝐴𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦𝐴) ∧ 𝑧 ∈ ω) ∧ (inl‘𝑦) = (𝑓𝑧)) → (case(( I ↾ 𝐴), (1o × {𝑥}))‘(inl‘𝑦)) = (( I ↾ 𝐴)‘𝑦))
3726, 28, 363eqtr2d 2216 . . . . . . . . . . . . 13 (((((𝑥𝐴𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦𝐴) ∧ 𝑧 ∈ ω) ∧ (inl‘𝑦) = (𝑓𝑧)) → ((case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓)‘𝑧) = (( I ↾ 𝐴)‘𝑦))
38 fvresi 5705 . . . . . . . . . . . . . 14 (𝑦𝐴 → (( I ↾ 𝐴)‘𝑦) = 𝑦)
3935, 38syl 14 . . . . . . . . . . . . 13 (((((𝑥𝐴𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦𝐴) ∧ 𝑧 ∈ ω) ∧ (inl‘𝑦) = (𝑓𝑧)) → (( I ↾ 𝐴)‘𝑦) = 𝑦)
4037, 39eqtr2d 2211 . . . . . . . . . . . 12 (((((𝑥𝐴𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦𝐴) ∧ 𝑧 ∈ ω) ∧ (inl‘𝑦) = (𝑓𝑧)) → 𝑦 = ((case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓)‘𝑧))
4140ex 115 . . . . . . . . . . 11 ((((𝑥𝐴𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦𝐴) ∧ 𝑧 ∈ ω) → ((inl‘𝑦) = (𝑓𝑧) → 𝑦 = ((case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓)‘𝑧)))
4241reximdva 2579 . . . . . . . . . 10 (((𝑥𝐴𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦𝐴) → (∃𝑧 ∈ ω (inl‘𝑦) = (𝑓𝑧) → ∃𝑧 ∈ ω 𝑦 = ((case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓)‘𝑧)))
4321, 42mpd 13 . . . . . . . . 9 (((𝑥𝐴𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦𝐴) → ∃𝑧 ∈ ω 𝑦 = ((case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓)‘𝑧))
4443ralrimiva 2550 . . . . . . . 8 ((𝑥𝐴𝑓:ω–onto→(𝐴 ⊔ 1o)) → ∀𝑦𝐴𝑧 ∈ ω 𝑦 = ((case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓)‘𝑧))
45 dffo3 5659 . . . . . . . 8 ((case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓):ω–onto𝐴 ↔ ((case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓):ω⟶𝐴 ∧ ∀𝑦𝐴𝑧 ∈ ω 𝑦 = ((case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓)‘𝑧)))
4616, 44, 45sylanbrc 417 . . . . . . 7 ((𝑥𝐴𝑓:ω–onto→(𝐴 ⊔ 1o)) → (case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓):ω–onto𝐴)
47 foeq1 5430 . . . . . . . 8 (𝑔 = (case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓) → (𝑔:ω–onto𝐴 ↔ (case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓):ω–onto𝐴))
4847spcegv 2825 . . . . . . 7 ((case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓) ∈ V → ((case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓):ω–onto𝐴 → ∃𝑔 𝑔:ω–onto𝐴))
4912, 46, 48sylc 62 . . . . . 6 ((𝑥𝐴𝑓:ω–onto→(𝐴 ⊔ 1o)) → ∃𝑔 𝑔:ω–onto𝐴)
5049ex 115 . . . . 5 (𝑥𝐴 → (𝑓:ω–onto→(𝐴 ⊔ 1o) → ∃𝑔 𝑔:ω–onto𝐴))
5150exlimiv 1598 . . . 4 (∃𝑥 𝑥𝐴 → (𝑓:ω–onto→(𝐴 ⊔ 1o) → ∃𝑔 𝑔:ω–onto𝐴))
5251exlimdv 1819 . . 3 (∃𝑥 𝑥𝐴 → (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) → ∃𝑔 𝑔:ω–onto𝐴))
53 foeq1 5430 . . . 4 (𝑓 = 𝑔 → (𝑓:ω–onto𝐴𝑔:ω–onto𝐴))
5453cbvexv 1918 . . 3 (∃𝑓 𝑓:ω–onto𝐴 ↔ ∃𝑔 𝑔:ω–onto𝐴)
5552, 54syl6ibr 162 . 2 (∃𝑥 𝑥𝐴 → (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) → ∃𝑓 𝑓:ω–onto𝐴))
56 ctmlemr 7101 . 2 (∃𝑥 𝑥𝐴 → (∃𝑓 𝑓:ω–onto𝐴 → ∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o)))
5755, 56impbid 129 1 (∃𝑥 𝑥𝐴 → (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) ↔ ∃𝑓 𝑓:ω–onto𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wex 1492  wcel 2148  wral 2455  wrex 2456  Vcvv 2737  {csn 3591   I cid 4285  ωcom 4586   × cxp 4621  cres 4625  ccom 4627  Fun wfun 5206   Fn wfn 5207  wf 5208  ontowfo 5210  1-1-ontowf1o 5211  cfv 5212  1oc1o 6404  cdju 7030  inlcinl 7038  casecdjucase 7076
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-if 3535  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  df-case 7077
This theorem is referenced by:  ctssdc  7106  enumct  7108  omct  7110  unbendc  12438  pw1nct  14408
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