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Theorem ccfunen 7072
 Description: Existence of a choice function for a countably infinite set. (Contributed by Jim Kingdon, 28-Nov-2023.)
Hypotheses
Ref Expression
ccfunen.cc (𝜑CCHOICE)
ccfunen.a (𝜑𝐴 ≈ ω)
ccfunen.m (𝜑 → ∀𝑥𝐴𝑤 𝑤𝑥)
Assertion
Ref Expression
ccfunen (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑥))
Distinct variable groups:   𝐴,𝑓,𝑥   𝜑,𝑓,𝑥   𝑥,𝑤
Allowed substitution hints:   𝜑(𝑤)   𝐴(𝑤)

Proof of Theorem ccfunen
Dummy variables 𝑢 𝑣 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ccfunen.a . . . . . 6 (𝜑𝐴 ≈ ω)
2 encv 6633 . . . . . 6 (𝐴 ≈ ω → (𝐴 ∈ V ∧ ω ∈ V))
31, 2syl 14 . . . . 5 (𝜑 → (𝐴 ∈ V ∧ ω ∈ V))
43simpld 111 . . . 4 (𝜑𝐴 ∈ V)
5 abid2 2258 . . . . . 6 {𝑣𝑣𝑢} = 𝑢
6 vex 2684 . . . . . 6 𝑢 ∈ V
75, 6eqeltri 2210 . . . . 5 {𝑣𝑣𝑢} ∈ V
87a1i 9 . . . 4 ((𝜑𝑢𝐴) → {𝑣𝑣𝑢} ∈ V)
94, 8opabex3d 6012 . . 3 (𝜑 → {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∈ V)
10 ccfunen.cc . . . 4 (𝜑CCHOICE)
11 df-cc 7071 . . . 4 (CCHOICE ↔ ∀𝑦(dom 𝑦 ≈ ω → ∃𝑓(𝑓𝑦𝑓 Fn dom 𝑦)))
1210, 11sylib 121 . . 3 (𝜑 → ∀𝑦(dom 𝑦 ≈ ω → ∃𝑓(𝑓𝑦𝑓 Fn dom 𝑦)))
13 ccfunen.m . . . . . 6 (𝜑 → ∀𝑥𝐴𝑤 𝑤𝑥)
14 elequ2 1691 . . . . . . . . 9 (𝑥 = 𝑢 → (𝑤𝑥𝑤𝑢))
1514exbidv 1797 . . . . . . . 8 (𝑥 = 𝑢 → (∃𝑤 𝑤𝑥 ↔ ∃𝑤 𝑤𝑢))
1615cbvralv 2652 . . . . . . 7 (∀𝑥𝐴𝑤 𝑤𝑥 ↔ ∀𝑢𝐴𝑤 𝑤𝑢)
17 elequ1 1690 . . . . . . . . 9 (𝑤 = 𝑣 → (𝑤𝑢𝑣𝑢))
1817cbvexv 1890 . . . . . . . 8 (∃𝑤 𝑤𝑢 ↔ ∃𝑣 𝑣𝑢)
1918ralbii 2439 . . . . . . 7 (∀𝑢𝐴𝑤 𝑤𝑢 ↔ ∀𝑢𝐴𝑣 𝑣𝑢)
2016, 19bitri 183 . . . . . 6 (∀𝑥𝐴𝑤 𝑤𝑥 ↔ ∀𝑢𝐴𝑣 𝑣𝑢)
2113, 20sylib 121 . . . . 5 (𝜑 → ∀𝑢𝐴𝑣 𝑣𝑢)
22 dmopab3 4747 . . . . 5 (∀𝑢𝐴𝑣 𝑣𝑢 ↔ dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} = 𝐴)
2321, 22sylib 121 . . . 4 (𝜑 → dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} = 𝐴)
2423, 1eqbrtrd 3945 . . 3 (𝜑 → dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ≈ ω)
25 dmeq 4734 . . . . . 6 (𝑦 = {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} → dom 𝑦 = dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})
2625breq1d 3934 . . . . 5 (𝑦 = {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} → (dom 𝑦 ≈ ω ↔ dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ≈ ω))
27 sseq2 3116 . . . . . . 7 (𝑦 = {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} → (𝑓𝑦𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)}))
2825fneq2d 5209 . . . . . . 7 (𝑦 = {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} → (𝑓 Fn dom 𝑦𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)}))
2927, 28anbi12d 464 . . . . . 6 (𝑦 = {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} → ((𝑓𝑦𝑓 Fn dom 𝑦) ↔ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})))
3029exbidv 1797 . . . . 5 (𝑦 = {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} → (∃𝑓(𝑓𝑦𝑓 Fn dom 𝑦) ↔ ∃𝑓(𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})))
3126, 30imbi12d 233 . . . 4 (𝑦 = {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} → ((dom 𝑦 ≈ ω → ∃𝑓(𝑓𝑦𝑓 Fn dom 𝑦)) ↔ (dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ≈ ω → ∃𝑓(𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)}))))
3231spcgv 2768 . . 3 ({⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∈ V → (∀𝑦(dom 𝑦 ≈ ω → ∃𝑓(𝑓𝑦𝑓 Fn dom 𝑦)) → (dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ≈ ω → ∃𝑓(𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)}))))
339, 12, 24, 32syl3c 63 . 2 (𝜑 → ∃𝑓(𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)}))
34 simprr 521 . . . . . 6 ((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})) → 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})
3523fneq2d 5209 . . . . . . 7 (𝜑 → (𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ↔ 𝑓 Fn 𝐴))
3635adantr 274 . . . . . 6 ((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})) → (𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ↔ 𝑓 Fn 𝐴))
3734, 36mpbid 146 . . . . 5 ((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})) → 𝑓 Fn 𝐴)
38 simplrl 524 . . . . . . . . 9 (((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})) ∧ 𝑥𝐴) → 𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})
39 fnopfv 5543 . . . . . . . . . 10 ((𝑓 Fn 𝐴𝑥𝐴) → ⟨𝑥, (𝑓𝑥)⟩ ∈ 𝑓)
4037, 39sylan 281 . . . . . . . . 9 (((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})) ∧ 𝑥𝐴) → ⟨𝑥, (𝑓𝑥)⟩ ∈ 𝑓)
4138, 40sseldd 3093 . . . . . . . 8 (((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})) ∧ 𝑥𝐴) → ⟨𝑥, (𝑓𝑥)⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})
42 vex 2684 . . . . . . . . 9 𝑥 ∈ V
43 vex 2684 . . . . . . . . . 10 𝑓 ∈ V
4443, 42fvex 5434 . . . . . . . . 9 (𝑓𝑥) ∈ V
45 eleq1 2200 . . . . . . . . . 10 (𝑢 = 𝑥 → (𝑢𝐴𝑥𝐴))
46 elequ2 1691 . . . . . . . . . 10 (𝑢 = 𝑥 → (𝑣𝑢𝑣𝑥))
4745, 46anbi12d 464 . . . . . . . . 9 (𝑢 = 𝑥 → ((𝑢𝐴𝑣𝑢) ↔ (𝑥𝐴𝑣𝑥)))
48 eleq1 2200 . . . . . . . . . 10 (𝑣 = (𝑓𝑥) → (𝑣𝑥 ↔ (𝑓𝑥) ∈ 𝑥))
4948anbi2d 459 . . . . . . . . 9 (𝑣 = (𝑓𝑥) → ((𝑥𝐴𝑣𝑥) ↔ (𝑥𝐴 ∧ (𝑓𝑥) ∈ 𝑥)))
5042, 44, 47, 49opelopab 4188 . . . . . . . 8 (⟨𝑥, (𝑓𝑥)⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ↔ (𝑥𝐴 ∧ (𝑓𝑥) ∈ 𝑥))
5141, 50sylib 121 . . . . . . 7 (((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})) ∧ 𝑥𝐴) → (𝑥𝐴 ∧ (𝑓𝑥) ∈ 𝑥))
5251simprd 113 . . . . . 6 (((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})) ∧ 𝑥𝐴) → (𝑓𝑥) ∈ 𝑥)
5352ralrimiva 2503 . . . . 5 ((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})) → ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑥)
5437, 53jca 304 . . . 4 ((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})) → (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑥))
5554ex 114 . . 3 (𝜑 → ((𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)}) → (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑥)))
5655eximdv 1852 . 2 (𝜑 → (∃𝑓(𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)}) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑥)))
5733, 56mpd 13 1 (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑥))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1329   = wceq 1331  ∃wex 1468   ∈ wcel 1480  {cab 2123  ∀wral 2414  Vcvv 2681   ⊆ wss 3066  ⟨cop 3525   class class class wbr 3924  {copab 3983  ωcom 4499  dom cdm 4534   Fn wfn 5113  ‘cfv 5118   ≈ cen 6625  CCHOICEwacc 7070 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 2119  ax-coll 4038  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-en 6628  df-cc 7071 This theorem is referenced by: (None)
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