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Theorem ctm 7125
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 5513 . . . . . . . . . . 11 ( I ↾ 𝐴):𝐴1-1-onto𝐴
2 f1of 5475 . . . . . . . . . . 11 (( I ↾ 𝐴):𝐴1-1-onto𝐴 → ( I ↾ 𝐴):𝐴𝐴)
31, 2mp1i 10 . . . . . . . . . 10 ((𝑥𝐴𝑓:ω–onto→(𝐴 ⊔ 1o)) → ( I ↾ 𝐴):𝐴𝐴)
4 fconst6g 5428 . . . . . . . . . . 11 (𝑥𝐴 → (1o × {𝑥}):1o𝐴)
54adantr 276 . . . . . . . . . 10 ((𝑥𝐴𝑓:ω–onto→(𝐴 ⊔ 1o)) → (1o × {𝑥}):1o𝐴)
63, 5casef 7104 . . . . . . . . 9 ((𝑥𝐴𝑓:ω–onto→(𝐴 ⊔ 1o)) → case(( I ↾ 𝐴), (1o × {𝑥})):(𝐴 ⊔ 1o)⟶𝐴)
7 ffun 5382 . . . . . . . . 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 2754 . . . . . . . . 9 𝑓 ∈ V
109a1i 9 . . . . . . . 8 ((𝑥𝐴𝑓:ω–onto→(𝐴 ⊔ 1o)) → 𝑓 ∈ V)
11 cofunexg 6127 . . . . . . . 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 5452 . . . . . . . . . 10 (𝑓:ω–onto→(𝐴 ⊔ 1o) → 𝑓:ω⟶(𝐴 ⊔ 1o))
1413adantl 277 . . . . . . . . 9 ((𝑥𝐴𝑓:ω–onto→(𝐴 ⊔ 1o)) → 𝑓:ω⟶(𝐴 ⊔ 1o))
15 fco 5395 . . . . . . . . 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 7067 . . . . . . . . . . . 12 (𝑦𝐴 → (inl‘𝑦) ∈ (𝐴 ⊔ 1o))
1918adantl 277 . . . . . . . . . . 11 (((𝑥𝐴𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦𝐴) → (inl‘𝑦) ∈ (𝐴 ⊔ 1o))
20 foelrn 5768 . . . . . . . . . . 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 5454 . . . . . . . . . . . . . . . 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 5600 . . . . . . . . . . . . . . 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 5533 . . . . . . . . . . . . . 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 5347 . . . . . . . . . . . . . . . 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 2754 . . . . . . . . . . . . . . . . 17 𝑥 ∈ V
3231fconst6 5429 . . . . . . . . . . . . . . . 16 (1o × {𝑥}):1o⟶V
33 ffun 5382 . . . . . . . . . . . . . . . 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 7107 . . . . . . . . . . . . . 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 2227 . . . . . . . . . . . . 13 (((((𝑥𝐴𝑓:ω–onto→(𝐴 ⊔ 1o)) ∧ 𝑦𝐴) ∧ 𝑧 ∈ ω) ∧ (inl‘𝑦) = (𝑓𝑧)) → ((case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓)‘𝑧) = (( I ↾ 𝐴)‘𝑦))
38 fvresi 5724 . . . . . . . . . . . . . 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 2222 . . . . . . . . . . . 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 2591 . . . . . . . . . 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 2562 . . . . . . . 8 ((𝑥𝐴𝑓:ω–onto→(𝐴 ⊔ 1o)) → ∀𝑦𝐴𝑧 ∈ ω 𝑦 = ((case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓)‘𝑧))
45 dffo3 5678 . . . . . . . 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 5448 . . . . . . . 8 (𝑔 = (case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓) → (𝑔:ω–onto𝐴 ↔ (case(( I ↾ 𝐴), (1o × {𝑥})) ∘ 𝑓):ω–onto𝐴))
4847spcegv 2839 . . . . . . 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 1608 . . . 4 (∃𝑥 𝑥𝐴 → (𝑓:ω–onto→(𝐴 ⊔ 1o) → ∃𝑔 𝑔:ω–onto𝐴))
5251exlimdv 1829 . . 3 (∃𝑥 𝑥𝐴 → (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) → ∃𝑔 𝑔:ω–onto𝐴))
53 foeq1 5448 . . . 4 (𝑓 = 𝑔 → (𝑓:ω–onto𝐴𝑔:ω–onto𝐴))
5453cbvexv 1929 . . 3 (∃𝑓 𝑓:ω–onto𝐴 ↔ ∃𝑔 𝑔:ω–onto𝐴)
5552, 54imbitrrdi 162 . 2 (∃𝑥 𝑥𝐴 → (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) → ∃𝑓 𝑓:ω–onto𝐴))
56 ctmlemr 7124 . 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 1363  wex 1502  wcel 2159  wral 2467  wrex 2468  Vcvv 2751  {csn 3606   I cid 4302  ωcom 4603   × cxp 4638  cres 4642  ccom 4644  Fun wfun 5224   Fn wfn 5225  wf 5226  ontowfo 5228  1-1-ontowf1o 5229  cfv 5230  1oc1o 6427  cdju 7053  inlcinl 7061  casecdjucase 7099
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-coll 4132  ax-sep 4135  ax-nul 4143  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-iinf 4601
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-ral 2472  df-rex 2473  df-reu 2474  df-rab 2476  df-v 2753  df-sbc 2977  df-csb 3072  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-nul 3437  df-if 3549  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-int 3859  df-iun 3902  df-br 4018  df-opab 4079  df-mpt 4080  df-tr 4116  df-id 4307  df-iord 4380  df-on 4382  df-suc 4385  df-iom 4604  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-f1 5235  df-fo 5236  df-f1o 5237  df-fv 5238  df-1st 6158  df-2nd 6159  df-1o 6434  df-dju 7054  df-inl 7063  df-inr 7064  df-case 7100
This theorem is referenced by:  ctssdc  7129  enumct  7131  omct  7133  unbendc  12472  pw1nct  15136
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