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Theorem cc4 7286
Description: Countable choice by showing the existence of a function 𝑓 which can choose a value at each index 𝑛 such that 𝜒 holds. (Contributed by Mario Carneiro, 7-Apr-2013.) (Revised by Jim Kingdon, 1-May-2024.)
Hypotheses
Ref Expression
cc4.cc (𝜑CCHOICE)
cc4.1 (𝜑𝐴𝑉)
cc4.2 (𝜑𝑁 ≈ ω)
cc4.3 (𝑥 = (𝑓𝑛) → (𝜓𝜒))
cc4.m (𝜑 → ∀𝑛𝑁𝑥𝐴 𝜓)
Assertion
Ref Expression
cc4 (𝜑 → ∃𝑓(𝑓:𝑁𝐴 ∧ ∀𝑛𝑁 𝜒))
Distinct variable groups:   𝐴,𝑓,𝑛,𝑥   𝑓,𝑁,𝑛   𝜒,𝑥   𝜑,𝑓,𝑛   𝜓,𝑓
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑛)   𝜒(𝑓,𝑛)   𝑁(𝑥)   𝑉(𝑥,𝑓,𝑛)

Proof of Theorem cc4
StepHypRef Expression
1 cc4.cc . 2 (𝜑CCHOICE)
2 cc4.1 . 2 (𝜑𝐴𝑉)
3 nfcv 2331 . 2 𝑛𝐴
4 cc4.2 . 2 (𝜑𝑁 ≈ ω)
5 cc4.3 . 2 (𝑥 = (𝑓𝑛) → (𝜓𝜒))
6 cc4.m . 2 (𝜑 → ∀𝑛𝑁𝑥𝐴 𝜓)
71, 2, 3, 4, 5, 6cc4f 7285 1 (𝜑 → ∃𝑓(𝑓:𝑁𝐴 ∧ ∀𝑛𝑁 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1363  wex 1502  wcel 2159  wral 2467  wrex 2468   class class class wbr 4017  ωcom 4603  wf 5226  cfv 5230  cen 6755  CCHOICEwacc 7278
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 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-pow 4188  ax-pr 4223  ax-un 4447  ax-iinf 4601
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  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-ral 2472  df-rex 2473  df-reu 2474  df-rab 2476  df-v 2753  df-sbc 2977  df-csb 3072  df-un 3147  df-in 3149  df-ss 3156  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-id 4307  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-2nd 6159  df-er 6552  df-en 6758  df-cc 7279
This theorem is referenced by: (None)
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