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Theorem cc4 7173
 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 2299 . 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 7172 1 (𝜑 → ∃𝑓(𝑓:𝑁𝐴 ∧ ∀𝑛𝑁 𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1335  ∃wex 1472   ∈ wcel 2128  ∀wral 2435  ∃wrex 2436   class class class wbr 3965  ωcom 4547  ⟶wf 5163  ‘cfv 5167   ≈ cen 6676  CCHOICEwacc 7165 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-pow 4134  ax-pr 4168  ax-un 4392  ax-iinf 4545 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4252  df-iom 4548  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-f1 5172  df-fo 5173  df-f1o 5174  df-fv 5175  df-2nd 6083  df-er 6473  df-en 6679  df-cc 7166 This theorem is referenced by: (None)
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