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Theorem cc1 7264
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 109 . . . . . 6 ((CCHOICE ∧ (𝑥 ≈ ω ∧ ∀𝑧𝑥𝑤 𝑤𝑧)) → CCHOICE)
2 simprl 529 . . . . . 6 ((CCHOICE ∧ (𝑥 ≈ ω ∧ ∀𝑧𝑥𝑤 𝑤𝑧)) → 𝑥 ≈ ω)
3 simprr 531 . . . . . . 7 ((CCHOICE ∧ (𝑥 ≈ ω ∧ ∀𝑧𝑥𝑤 𝑤𝑧)) → ∀𝑧𝑥𝑤 𝑤𝑧)
4 elequ2 2153 . . . . . . . . 9 (𝑧 = 𝑎 → (𝑤𝑧𝑤𝑎))
54exbidv 1825 . . . . . . . 8 (𝑧 = 𝑎 → (∃𝑤 𝑤𝑧 ↔ ∃𝑤 𝑤𝑎))
65cbvralvw 2708 . . . . . . 7 (∀𝑧𝑥𝑤 𝑤𝑧 ↔ ∀𝑎𝑥𝑤 𝑤𝑎)
73, 6sylib 122 . . . . . 6 ((CCHOICE ∧ (𝑥 ≈ ω ∧ ∀𝑧𝑥𝑤 𝑤𝑧)) → ∀𝑎𝑥𝑤 𝑤𝑎)
81, 2, 7ccfunen 7263 . . . . 5 ((CCHOICE ∧ (𝑥 ≈ ω ∧ ∀𝑧𝑥𝑤 𝑤𝑧)) → ∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑎𝑥 (𝑓𝑎) ∈ 𝑎))
9 exsimpr 1618 . . . . 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 5516 . . . . . . 7 (𝑎 = 𝑧 → (𝑓𝑎) = (𝑓𝑧))
12 id 19 . . . . . . 7 (𝑎 = 𝑧𝑎 = 𝑧)
1311, 12eleq12d 2248 . . . . . 6 (𝑎 = 𝑧 → ((𝑓𝑎) ∈ 𝑎 ↔ (𝑓𝑧) ∈ 𝑧))
1413cbvralvw 2708 . . . . 5 (∀𝑎𝑥 (𝑓𝑎) ∈ 𝑎 ↔ ∀𝑧𝑥 (𝑓𝑧) ∈ 𝑧)
1514exbii 1605 . . . 4 (∃𝑓𝑎𝑥 (𝑓𝑎) ∈ 𝑎 ↔ ∃𝑓𝑧𝑥 (𝑓𝑧) ∈ 𝑧)
1610, 15sylib 122 . . 3 ((CCHOICE ∧ (𝑥 ≈ ω ∧ ∀𝑧𝑥𝑤 𝑤𝑧)) → ∃𝑓𝑧𝑥 (𝑓𝑧) ∈ 𝑧)
1716ex 115 . 2 (CCHOICE → ((𝑥 ≈ ω ∧ ∀𝑧𝑥𝑤 𝑤𝑧) → ∃𝑓𝑧𝑥 (𝑓𝑧) ∈ 𝑧))
1817alrimiv 1874 1 (CCHOICE → ∀𝑥((𝑥 ≈ ω ∧ ∀𝑧𝑥𝑤 𝑤𝑧) → ∃𝑓𝑧𝑥 (𝑓𝑧) ∈ 𝑧))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wal 1351  wex 1492  wcel 2148  wral 2455   class class class wbr 4004  ωcom 4590   Fn wfn 5212  cfv 5217  cen 6738  CCHOICEwacc 7261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-en 6741  df-cc 7262
This theorem is referenced by: (None)
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