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Theorem cc4n 7299
Description: Countable choice with a simpler restriction on how every set in the countable collection needs to be inhabited. That is, compared with cc4 7298, the hypotheses only require an A(n) for each value of 𝑛, not a single set 𝐴 which suffices for every 𝑛 ∈ ω. (Contributed by Mario Carneiro, 7-Apr-2013.) (Revised by Jim Kingdon, 3-May-2024.)
Hypotheses
Ref Expression
cc4n.cc (𝜑CCHOICE)
cc4n.1 (𝜑 → ∀𝑛𝑁 {𝑥𝐴𝜓} ∈ 𝑉)
cc4n.2 (𝜑𝑁 ≈ ω)
cc4n.3 (𝑥 = (𝑓𝑛) → (𝜓𝜒))
cc4n.m (𝜑 → ∀𝑛𝑁𝑥𝐴 𝜓)
Assertion
Ref Expression
cc4n (𝜑 → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 𝜒))
Distinct variable groups:   𝐴,𝑓,𝑥   𝑓,𝑁,𝑛   𝜒,𝑥   𝜑,𝑓,𝑛   𝜓,𝑓   𝑥,𝑛
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑛)   𝜒(𝑓,𝑛)   𝐴(𝑛)   𝑁(𝑥)   𝑉(𝑥,𝑓,𝑛)

Proof of Theorem cc4n
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 cc4n.cc . . 3 (𝜑CCHOICE)
2 cc4n.1 . . . 4 (𝜑 → ∀𝑛𝑁 {𝑥𝐴𝜓} ∈ 𝑉)
3 elex 2763 . . . . 5 ({𝑥𝐴𝜓} ∈ 𝑉 → {𝑥𝐴𝜓} ∈ V)
43ralimi 2553 . . . 4 (∀𝑛𝑁 {𝑥𝐴𝜓} ∈ 𝑉 → ∀𝑛𝑁 {𝑥𝐴𝜓} ∈ V)
52, 4syl 14 . . 3 (𝜑 → ∀𝑛𝑁 {𝑥𝐴𝜓} ∈ V)
6 cc4n.m . . . 4 (𝜑 → ∀𝑛𝑁𝑥𝐴 𝜓)
7 rabn0m 3465 . . . . 5 (∃𝑤 𝑤 ∈ {𝑥𝐴𝜓} ↔ ∃𝑥𝐴 𝜓)
87ralbii 2496 . . . 4 (∀𝑛𝑁𝑤 𝑤 ∈ {𝑥𝐴𝜓} ↔ ∀𝑛𝑁𝑥𝐴 𝜓)
96, 8sylibr 134 . . 3 (𝜑 → ∀𝑛𝑁𝑤 𝑤 ∈ {𝑥𝐴𝜓})
10 cc4n.2 . . 3 (𝜑𝑁 ≈ ω)
111, 5, 9, 10cc3 7296 . 2 (𝜑 → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝑓𝑛) ∈ {𝑥𝐴𝜓}))
12 simprl 529 . . . . 5 ((𝜑 ∧ (𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝑓𝑛) ∈ {𝑥𝐴𝜓})) → 𝑓 Fn 𝑁)
13 cc4n.3 . . . . . . . . 9 (𝑥 = (𝑓𝑛) → (𝜓𝜒))
1413elrab 2908 . . . . . . . 8 ((𝑓𝑛) ∈ {𝑥𝐴𝜓} ↔ ((𝑓𝑛) ∈ 𝐴𝜒))
1514simprbi 275 . . . . . . 7 ((𝑓𝑛) ∈ {𝑥𝐴𝜓} → 𝜒)
1615ralimi 2553 . . . . . 6 (∀𝑛𝑁 (𝑓𝑛) ∈ {𝑥𝐴𝜓} → ∀𝑛𝑁 𝜒)
1716ad2antll 491 . . . . 5 ((𝜑 ∧ (𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝑓𝑛) ∈ {𝑥𝐴𝜓})) → ∀𝑛𝑁 𝜒)
1812, 17jca 306 . . . 4 ((𝜑 ∧ (𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝑓𝑛) ∈ {𝑥𝐴𝜓})) → (𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 𝜒))
1918ex 115 . . 3 (𝜑 → ((𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝑓𝑛) ∈ {𝑥𝐴𝜓}) → (𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 𝜒)))
2019eximdv 1891 . 2 (𝜑 → (∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝑓𝑛) ∈ {𝑥𝐴𝜓}) → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 𝜒)))
2111, 20mpd 13 1 (𝜑 → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wex 1503  wcel 2160  wral 2468  wrex 2469  {crab 2472  Vcvv 2752   class class class wbr 4018  ωcom 4607   Fn wfn 5230  cfv 5235  cen 6763  CCHOICEwacc 7290
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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-2nd 6165  df-er 6558  df-en 6766  df-cc 7291
This theorem is referenced by:  omctfn  12493
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