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Theorem cc4n 7091
 Description: Countable choice with a simpler restriction on how every set in the countable collection needs to be inhabited. That is, compared with cc4 7090, 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 2697 . . . . 5 ({𝑥𝐴𝜓} ∈ 𝑉 → {𝑥𝐴𝜓} ∈ V)
43ralimi 2495 . . . 4 (∀𝑛𝑁 {𝑥𝐴𝜓} ∈ 𝑉 → ∀𝑛𝑁 {𝑥𝐴𝜓} ∈ V)
52, 4syl 14 . . 3 (𝜑 → ∀𝑛𝑁 {𝑥𝐴𝜓} ∈ V)
6 cc4n.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 cc4n.2 . . 3 (𝜑𝑁 ≈ ω)
111, 5, 9, 10cc3 7088 . 2 (𝜑 → ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝑓𝑛) ∈ {𝑥𝐴𝜓}))
12 simprl 520 . . . . 5 ((𝜑 ∧ (𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝑓𝑛) ∈ {𝑥𝐴𝜓})) → 𝑓 Fn 𝑁)
13 cc4n.3 . . . . . . . . 9 (𝑥 = (𝑓𝑛) → (𝜓𝜒))
1413elrab 2840 . . . . . . . 8 ((𝑓𝑛) ∈ {𝑥𝐴𝜓} ↔ ((𝑓𝑛) ∈ 𝐴𝜒))
1514simprbi 273 . . . . . . 7 ((𝑓𝑛) ∈ {𝑥𝐴𝜓} → 𝜒)
1615ralimi 2495 . . . . . 6 (∀𝑛𝑁 (𝑓𝑛) ∈ {𝑥𝐴𝜓} → ∀𝑛𝑁 𝜒)
1716ad2antll 482 . . . . 5 ((𝜑 ∧ (𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝑓𝑛) ∈ {𝑥𝐴𝜓})) → ∀𝑛𝑁 𝜒)
1812, 17jca 304 . . . 4 ((𝜑 ∧ (𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝑓𝑛) ∈ {𝑥𝐴𝜓})) → (𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 𝜒))
1918ex 114 . . 3 (𝜑 → ((𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 (𝑓𝑛) ∈ {𝑥𝐴𝜓}) → (𝑓 Fn 𝑁 ∧ ∀𝑛𝑁 𝜒)))
2019eximdv 1852 . 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 103   ↔ wb 104   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∀wral 2416  ∃wrex 2417  {crab 2420  Vcvv 2686   class class class wbr 3929  ωcom 4504   Fn wfn 5118  ‘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:  omctfn  11967
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