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Theorem cc2 7098
Description: Countable choice using sequences instead of countable sets. (Contributed by Jim Kingdon, 27-Apr-2024.)
Hypotheses
Ref Expression
cc2.cc (𝜑CCHOICE)
cc2.a (𝜑𝐹 Fn ω)
cc2.m (𝜑 → ∀𝑥 ∈ ω ∃𝑤 𝑤 ∈ (𝐹𝑥))
Assertion
Ref Expression
cc2 (𝜑 → ∃𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω (𝑔𝑛) ∈ (𝐹𝑛)))
Distinct variable groups:   𝑔,𝐹,𝑛   𝑤,𝐹,𝑥   𝜑,𝑛
Allowed substitution hints:   𝜑(𝑥,𝑤,𝑔)

Proof of Theorem cc2
Dummy variables 𝑓 𝑚 𝑣 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cc2.cc . 2 (𝜑CCHOICE)
2 cc2.a . 2 (𝜑𝐹 Fn ω)
3 cc2.m . . . 4 (𝜑 → ∀𝑥 ∈ ω ∃𝑤 𝑤 ∈ (𝐹𝑥))
4 fveq2 5428 . . . . . . 7 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
54eleq2d 2210 . . . . . 6 (𝑥 = 𝑦 → (𝑤 ∈ (𝐹𝑥) ↔ 𝑤 ∈ (𝐹𝑦)))
65exbidv 1798 . . . . 5 (𝑥 = 𝑦 → (∃𝑤 𝑤 ∈ (𝐹𝑥) ↔ ∃𝑤 𝑤 ∈ (𝐹𝑦)))
76cbvralv 2657 . . . 4 (∀𝑥 ∈ ω ∃𝑤 𝑤 ∈ (𝐹𝑥) ↔ ∀𝑦 ∈ ω ∃𝑤 𝑤 ∈ (𝐹𝑦))
83, 7sylib 121 . . 3 (𝜑 → ∀𝑦 ∈ ω ∃𝑤 𝑤 ∈ (𝐹𝑦))
9 eleq1w 2201 . . . . 5 (𝑤 = 𝑣 → (𝑤 ∈ (𝐹𝑦) ↔ 𝑣 ∈ (𝐹𝑦)))
109cbvexv 1891 . . . 4 (∃𝑤 𝑤 ∈ (𝐹𝑦) ↔ ∃𝑣 𝑣 ∈ (𝐹𝑦))
1110ralbii 2444 . . 3 (∀𝑦 ∈ ω ∃𝑤 𝑤 ∈ (𝐹𝑦) ↔ ∀𝑦 ∈ ω ∃𝑣 𝑣 ∈ (𝐹𝑦))
128, 11sylib 121 . 2 (𝜑 → ∀𝑦 ∈ ω ∃𝑣 𝑣 ∈ (𝐹𝑦))
13 nfcv 2282 . . 3 𝑛({𝑚} × (𝐹𝑚))
14 nfcv 2282 . . 3 𝑚({𝑛} × (𝐹𝑛))
15 sneq 3542 . . . 4 (𝑚 = 𝑛 → {𝑚} = {𝑛})
16 fveq2 5428 . . . 4 (𝑚 = 𝑛 → (𝐹𝑚) = (𝐹𝑛))
1715, 16xpeq12d 4571 . . 3 (𝑚 = 𝑛 → ({𝑚} × (𝐹𝑚)) = ({𝑛} × (𝐹𝑛)))
1813, 14, 17cbvmpt 4030 . 2 (𝑚 ∈ ω ↦ ({𝑚} × (𝐹𝑚))) = (𝑛 ∈ ω ↦ ({𝑛} × (𝐹𝑛)))
19 nfcv 2282 . . 3 𝑛(2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × (𝐹𝑚)))‘𝑚)))
20 nfcv 2282 . . . 4 𝑚2nd
21 nfcv 2282 . . . . 5 𝑚𝑓
22 nffvmpt1 5439 . . . . 5 𝑚((𝑚 ∈ ω ↦ ({𝑚} × (𝐹𝑚)))‘𝑛)
2321, 22nffv 5438 . . . 4 𝑚(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × (𝐹𝑚)))‘𝑛))
2420, 23nffv 5438 . . 3 𝑚(2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × (𝐹𝑚)))‘𝑛)))
25 2fveq3 5433 . . . 4 (𝑚 = 𝑛 → (𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × (𝐹𝑚)))‘𝑚)) = (𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × (𝐹𝑚)))‘𝑛)))
2625fveq2d 5432 . . 3 (𝑚 = 𝑛 → (2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × (𝐹𝑚)))‘𝑚))) = (2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × (𝐹𝑚)))‘𝑛))))
2719, 24, 26cbvmpt 4030 . 2 (𝑚 ∈ ω ↦ (2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × (𝐹𝑚)))‘𝑚)))) = (𝑛 ∈ ω ↦ (2nd ‘(𝑓‘((𝑚 ∈ ω ↦ ({𝑚} × (𝐹𝑚)))‘𝑛))))
281, 2, 12, 18, 27cc2lem 7097 1 (𝜑 → ∃𝑔(𝑔 Fn ω ∧ ∀𝑛 ∈ ω (𝑔𝑛) ∈ (𝐹𝑛)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wex 1469  wcel 1481  wral 2417  {csn 3531  cmpt 3996  ωcom 4511   × cxp 4544   Fn wfn 5125  cfv 5130  2nd c2nd 6044  CCHOICEwacc 7093
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-iinf 4509
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-2nd 6046  df-er 6436  df-en 6642  df-cc 7094
This theorem is referenced by:  cc3  7099
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