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Theorem cc4f 7089
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, 3-May-2024.)
Hypotheses
Ref Expression
cc4f.cc (𝜑CCHOICE)
cc4f.1 (𝜑𝐴𝑉)
cc4f.a 𝑛𝐴
cc4f.2 (𝜑𝑁 ≈ ω)
cc4f.3 (𝑥 = (𝑓𝑛) → (𝜓𝜒))
cc4f.m (𝜑 → ∀𝑛𝑁𝑥𝐴 𝜓)
Assertion
Ref Expression
cc4f (𝜑 → ∃𝑓(𝑓:𝑁𝐴 ∧ ∀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 cc4f
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 cc4f.cc . . 3 (𝜑CCHOICE)
2 cc4f.1 . . . . 5 (𝜑𝐴𝑉)
3 rabexg 4071 . . . . 5 (𝐴𝑉 → {𝑥𝐴𝜓} ∈ V)
42, 3syl 14 . . . 4 (𝜑 → {𝑥𝐴𝜓} ∈ V)
54ralrimivw 2506 . . 3 (𝜑 → ∀𝑛𝑁 {𝑥𝐴𝜓} ∈ V)
6 cc4f.m . . . 4 (𝜑 → ∀𝑛𝑁𝑥𝐴 𝜓)
7 rabn0m 3390 . . . . 5 (∃𝑤 𝑤 ∈ {𝑥𝐴𝜓} ↔ ∃𝑥𝐴 𝜓)
87ralbii 2441 . . . 4 (∀𝑛𝑁𝑤 𝑤 ∈ {𝑥𝐴𝜓} ↔ ∀𝑛𝑁𝑥𝐴 𝜓)
96, 8sylibr 133 . . 3 (𝜑 → ∀𝑛𝑁𝑤 𝑤 ∈ {𝑥𝐴𝜓})
10 cc4f.2 . . 3 (𝜑𝑁 ≈ ω)
111, 5, 9, 10cc3 7088 . 2 (𝜑 → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝑓𝑛) ∈ {𝑥𝐴𝜓}))
12 simprl 520 . . . . . 6 ((𝜑 ∧ (𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝑓𝑛) ∈ {𝑥𝐴𝜓})) → 𝑓 Fn 𝑁)
13 elrabi 2837 . . . . . . . 8 ((𝑓𝑛) ∈ {𝑥𝐴𝜓} → (𝑓𝑛) ∈ 𝐴)
1413ralimi 2495 . . . . . . 7 (∀𝑛𝑁 (𝑓𝑛) ∈ {𝑥𝐴𝜓} → ∀𝑛𝑁 (𝑓𝑛) ∈ 𝐴)
1514ad2antll 482 . . . . . 6 ((𝜑 ∧ (𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝑓𝑛) ∈ {𝑥𝐴𝜓})) → ∀𝑛𝑁 (𝑓𝑛) ∈ 𝐴)
16 nfcv 2281 . . . . . . 7 𝑛𝑁
17 cc4f.a . . . . . . 7 𝑛𝐴
18 nfcv 2281 . . . . . . 7 𝑛𝑓
1916, 17, 18ffnfvf 5579 . . . . . 6 (𝑓:𝑁𝐴 ↔ (𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝑓𝑛) ∈ 𝐴))
2012, 15, 19sylanbrc 413 . . . . 5 ((𝜑 ∧ (𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝑓𝑛) ∈ {𝑥𝐴𝜓})) → 𝑓:𝑁𝐴)
21 cc4f.3 . . . . . . . . 9 (𝑥 = (𝑓𝑛) → (𝜓𝜒))
2221elrab 2840 . . . . . . . 8 ((𝑓𝑛) ∈ {𝑥𝐴𝜓} ↔ ((𝑓𝑛) ∈ 𝐴𝜒))
2322simprbi 273 . . . . . . 7 ((𝑓𝑛) ∈ {𝑥𝐴𝜓} → 𝜒)
2423ralimi 2495 . . . . . 6 (∀𝑛𝑁 (𝑓𝑛) ∈ {𝑥𝐴𝜓} → ∀𝑛𝑁 𝜒)
2524ad2antll 482 . . . . 5 ((𝜑 ∧ (𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝑓𝑛) ∈ {𝑥𝐴𝜓})) → ∀𝑛𝑁 𝜒)
2620, 25jca 304 . . . 4 ((𝜑 ∧ (𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝑓𝑛) ∈ {𝑥𝐴𝜓})) → (𝑓:𝑁𝐴 ∧ ∀𝑛𝑁 𝜒))
2726ex 114 . . 3 (𝜑 → ((𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝑓𝑛) ∈ {𝑥𝐴𝜓}) → (𝑓:𝑁𝐴 ∧ ∀𝑛𝑁 𝜒)))
2827eximdv 1852 . 2 (𝜑 → (∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝑓𝑛) ∈ {𝑥𝐴𝜓}) → ∃𝑓(𝑓:𝑁𝐴 ∧ ∀𝑛𝑁 𝜒)))
2911, 28mpd 13 1 (𝜑 → ∃𝑓(𝑓:𝑁𝐴 ∧ ∀𝑛𝑁 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wex 1468  wcel 1480  wnfc 2268  wral 2416  wrex 2417  {crab 2420  Vcvv 2686   class class class wbr 3929  ωcom 4504   Fn wfn 5118  wf 5119  cfv 5123  cen 6632  CCHOICEwacc 7082
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-2nd 6039  df-er 6429  df-en 6635  df-cc 7083
This theorem is referenced by:  cc4  7090
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