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Theorem cc1 7096
 Description: Countable choice in terms of a choice function on a countably infinite set of inhabited sets. (Contributed by Jim Kingdon, 27-Apr-2024.)
Assertion
Ref Expression
cc1 (CCHOICE → ∀𝑥((𝑥 ≈ ω ∧ ∀𝑧𝑥𝑤 𝑤𝑧) → ∃𝑓𝑧𝑥 (𝑓𝑧) ∈ 𝑧))
Distinct variable groups:   𝑤,𝑓,𝑧   𝑥,𝑓,𝑧

Proof of Theorem cc1
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 simpl 108 . . . . . 6 ((CCHOICE ∧ (𝑥 ≈ ω ∧ ∀𝑧𝑥𝑤 𝑤𝑧)) → CCHOICE)
2 simprl 521 . . . . . 6 ((CCHOICE ∧ (𝑥 ≈ ω ∧ ∀𝑧𝑥𝑤 𝑤𝑧)) → 𝑥 ≈ ω)
3 simprr 522 . . . . . . 7 ((CCHOICE ∧ (𝑥 ≈ ω ∧ ∀𝑧𝑥𝑤 𝑤𝑧)) → ∀𝑧𝑥𝑤 𝑤𝑧)
4 elequ2 1692 . . . . . . . . 9 (𝑧 = 𝑎 → (𝑤𝑧𝑤𝑎))
54exbidv 1798 . . . . . . . 8 (𝑧 = 𝑎 → (∃𝑤 𝑤𝑧 ↔ ∃𝑤 𝑤𝑎))
65cbvralvw 2661 . . . . . . 7 (∀𝑧𝑥𝑤 𝑤𝑧 ↔ ∀𝑎𝑥𝑤 𝑤𝑎)
73, 6sylib 121 . . . . . 6 ((CCHOICE ∧ (𝑥 ≈ ω ∧ ∀𝑧𝑥𝑤 𝑤𝑧)) → ∀𝑎𝑥𝑤 𝑤𝑎)
81, 2, 7ccfunen 7095 . . . . 5 ((CCHOICE ∧ (𝑥 ≈ ω ∧ ∀𝑧𝑥𝑤 𝑤𝑧)) → ∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑎𝑥 (𝑓𝑎) ∈ 𝑎))
9 exsimpr 1598 . . . . 5 (∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑎𝑥 (𝑓𝑎) ∈ 𝑎) → ∃𝑓𝑎𝑥 (𝑓𝑎) ∈ 𝑎)
108, 9syl 14 . . . 4 ((CCHOICE ∧ (𝑥 ≈ ω ∧ ∀𝑧𝑥𝑤 𝑤𝑧)) → ∃𝑓𝑎𝑥 (𝑓𝑎) ∈ 𝑎)
11 fveq2 5428 . . . . . . 7 (𝑎 = 𝑧 → (𝑓𝑎) = (𝑓𝑧))
12 id 19 . . . . . . 7 (𝑎 = 𝑧𝑎 = 𝑧)
1311, 12eleq12d 2211 . . . . . 6 (𝑎 = 𝑧 → ((𝑓𝑎) ∈ 𝑎 ↔ (𝑓𝑧) ∈ 𝑧))
1413cbvralvw 2661 . . . . 5 (∀𝑎𝑥 (𝑓𝑎) ∈ 𝑎 ↔ ∀𝑧𝑥 (𝑓𝑧) ∈ 𝑧)
1514exbii 1585 . . . 4 (∃𝑓𝑎𝑥 (𝑓𝑎) ∈ 𝑎 ↔ ∃𝑓𝑧𝑥 (𝑓𝑧) ∈ 𝑧)
1610, 15sylib 121 . . 3 ((CCHOICE ∧ (𝑥 ≈ ω ∧ ∀𝑧𝑥𝑤 𝑤𝑧)) → ∃𝑓𝑧𝑥 (𝑓𝑧) ∈ 𝑧)
1716ex 114 . 2 (CCHOICE → ((𝑥 ≈ ω ∧ ∀𝑧𝑥𝑤 𝑤𝑧) → ∃𝑓𝑧𝑥 (𝑓𝑧) ∈ 𝑧))
1817alrimiv 1847 1 (CCHOICE → ∀𝑥((𝑥 ≈ ω ∧ ∀𝑧𝑥𝑤 𝑤𝑧) → ∃𝑓𝑧𝑥 (𝑓𝑧) ∈ 𝑧))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1330  ∃wex 1469   ∈ wcel 1481  ∀wral 2417   class class class wbr 3936  ωcom 4511   Fn wfn 5125  ‘cfv 5130   ≈ cen 6639  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 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-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  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-en 6642  df-cc 7094 This theorem is referenced by: (None)
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