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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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