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Theorem ccfunen 7262
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 6745 . . . . . 6 (𝐴 ≈ ω → (𝐴 ∈ V ∧ ω ∈ V))
31, 2syl 14 . . . . 5 (𝜑 → (𝐴 ∈ V ∧ ω ∈ V))
43simpld 112 . . . 4 (𝜑𝐴 ∈ V)
5 abid2 2298 . . . . . 6 {𝑣𝑣𝑢} = 𝑢
6 vex 2740 . . . . . 6 𝑢 ∈ V
75, 6eqeltri 2250 . . . . 5 {𝑣𝑣𝑢} ∈ V
87a1i 9 . . . 4 ((𝜑𝑢𝐴) → {𝑣𝑣𝑢} ∈ V)
94, 8opabex3d 6121 . . 3 (𝜑 → {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∈ V)
10 ccfunen.cc . . . 4 (𝜑CCHOICE)
11 df-cc 7261 . . . 4 (CCHOICE ↔ ∀𝑦(dom 𝑦 ≈ ω → ∃𝑓(𝑓𝑦𝑓 Fn dom 𝑦)))
1210, 11sylib 122 . . 3 (𝜑 → ∀𝑦(dom 𝑦 ≈ ω → ∃𝑓(𝑓𝑦𝑓 Fn dom 𝑦)))
13 ccfunen.m . . . . . 6 (𝜑 → ∀𝑥𝐴𝑤 𝑤𝑥)
14 elequ2 2153 . . . . . . . . 9 (𝑥 = 𝑢 → (𝑤𝑥𝑤𝑢))
1514exbidv 1825 . . . . . . . 8 (𝑥 = 𝑢 → (∃𝑤 𝑤𝑥 ↔ ∃𝑤 𝑤𝑢))
1615cbvralv 2703 . . . . . . 7 (∀𝑥𝐴𝑤 𝑤𝑥 ↔ ∀𝑢𝐴𝑤 𝑤𝑢)
17 elequ1 2152 . . . . . . . . 9 (𝑤 = 𝑣 → (𝑤𝑢𝑣𝑢))
1817cbvexv 1918 . . . . . . . 8 (∃𝑤 𝑤𝑢 ↔ ∃𝑣 𝑣𝑢)
1918ralbii 2483 . . . . . . 7 (∀𝑢𝐴𝑤 𝑤𝑢 ↔ ∀𝑢𝐴𝑣 𝑣𝑢)
2016, 19bitri 184 . . . . . 6 (∀𝑥𝐴𝑤 𝑤𝑥 ↔ ∀𝑢𝐴𝑣 𝑣𝑢)
2113, 20sylib 122 . . . . 5 (𝜑 → ∀𝑢𝐴𝑣 𝑣𝑢)
22 dmopab3 4840 . . . . 5 (∀𝑢𝐴𝑣 𝑣𝑢 ↔ dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} = 𝐴)
2321, 22sylib 122 . . . 4 (𝜑 → dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} = 𝐴)
2423, 1eqbrtrd 4025 . . 3 (𝜑 → dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ≈ ω)
25 dmeq 4827 . . . . . 6 (𝑦 = {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} → dom 𝑦 = dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})
2625breq1d 4013 . . . . 5 (𝑦 = {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} → (dom 𝑦 ≈ ω ↔ dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ≈ ω))
27 sseq2 3179 . . . . . . 7 (𝑦 = {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} → (𝑓𝑦𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)}))
2825fneq2d 5307 . . . . . . 7 (𝑦 = {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} → (𝑓 Fn dom 𝑦𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)}))
2927, 28anbi12d 473 . . . . . 6 (𝑦 = {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} → ((𝑓𝑦𝑓 Fn dom 𝑦) ↔ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})))
3029exbidv 1825 . . . . 5 (𝑦 = {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} → (∃𝑓(𝑓𝑦𝑓 Fn dom 𝑦) ↔ ∃𝑓(𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})))
3126, 30imbi12d 234 . . . 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 2824 . . 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 531 . . . . . 6 ((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})) → 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})
3523fneq2d 5307 . . . . . . 7 (𝜑 → (𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ↔ 𝑓 Fn 𝐴))
3635adantr 276 . . . . . 6 ((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})) → (𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ↔ 𝑓 Fn 𝐴))
3734, 36mpbid 147 . . . . 5 ((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})) → 𝑓 Fn 𝐴)
38 simplrl 535 . . . . . . . . 9 (((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})) ∧ 𝑥𝐴) → 𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})
39 fnopfv 5646 . . . . . . . . . 10 ((𝑓 Fn 𝐴𝑥𝐴) → ⟨𝑥, (𝑓𝑥)⟩ ∈ 𝑓)
4037, 39sylan 283 . . . . . . . . 9 (((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})) ∧ 𝑥𝐴) → ⟨𝑥, (𝑓𝑥)⟩ ∈ 𝑓)
4138, 40sseldd 3156 . . . . . . . 8 (((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})) ∧ 𝑥𝐴) → ⟨𝑥, (𝑓𝑥)⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})
42 vex 2740 . . . . . . . . 9 𝑥 ∈ V
43 vex 2740 . . . . . . . . . 10 𝑓 ∈ V
4443, 42fvex 5535 . . . . . . . . 9 (𝑓𝑥) ∈ V
45 eleq1 2240 . . . . . . . . . 10 (𝑢 = 𝑥 → (𝑢𝐴𝑥𝐴))
46 elequ2 2153 . . . . . . . . . 10 (𝑢 = 𝑥 → (𝑣𝑢𝑣𝑥))
4745, 46anbi12d 473 . . . . . . . . 9 (𝑢 = 𝑥 → ((𝑢𝐴𝑣𝑢) ↔ (𝑥𝐴𝑣𝑥)))
48 eleq1 2240 . . . . . . . . . 10 (𝑣 = (𝑓𝑥) → (𝑣𝑥 ↔ (𝑓𝑥) ∈ 𝑥))
4948anbi2d 464 . . . . . . . . 9 (𝑣 = (𝑓𝑥) → ((𝑥𝐴𝑣𝑥) ↔ (𝑥𝐴 ∧ (𝑓𝑥) ∈ 𝑥)))
5042, 44, 47, 49opelopab 4271 . . . . . . . 8 (⟨𝑥, (𝑓𝑥)⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ↔ (𝑥𝐴 ∧ (𝑓𝑥) ∈ 𝑥))
5141, 50sylib 122 . . . . . . 7 (((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})) ∧ 𝑥𝐴) → (𝑥𝐴 ∧ (𝑓𝑥) ∈ 𝑥))
5251simprd 114 . . . . . 6 (((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})) ∧ 𝑥𝐴) → (𝑓𝑥) ∈ 𝑥)
5352ralrimiva 2550 . . . . 5 ((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})) → ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑥)
5437, 53jca 306 . . . 4 ((𝜑 ∧ (𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)})) → (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑥))
5554ex 115 . . 3 (𝜑 → ((𝑓 ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)} ∧ 𝑓 Fn dom {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣𝑢)}) → (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑥)))
5655eximdv 1880 . 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 104  wb 105  wal 1351   = wceq 1353  wex 1492  wcel 2148  {cab 2163  wral 2455  Vcvv 2737  wss 3129  cop 3595   class class class wbr 4003  {copab 4063  ωcom 4589  dom cdm 4626   Fn wfn 5211  cfv 5216  cen 6737  CCHOICEwacc 7260
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 4118  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433
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 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-en 6740  df-cc 7261
This theorem is referenced by:  cc1  7263  cc2lem  7264
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