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Theorem dfac11 39999
Description: The right-hand side of this theorem (compare with ac4 9890), sometimes known as the "axiom of multiple choice", is a choice equivalent. Curiously, this statement cannot be proved without ax-reg 9044, despite not mentioning the cumulative hierarchy in any way as most consequences of regularity do.

This is definition (MC) of [Schechter] p. 141. EDITORIAL: the proof is not original with me of course but I lost my reference sometime after writing it.

A multiple choice function allows any total order to be extended to a choice function, which in turn defines a well-ordering. Since a well-ordering on a set defines a simple ordering of the power set, this allows the trivial well-ordering of the empty set to be transfinitely bootstrapped up the cumulative hierarchy to any desired level. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Stefan O'Rear, 1-Jun-2015.)

Assertion
Ref Expression
dfac11 (CHOICE ↔ ∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
Distinct variable group:   𝑥,𝑧,𝑓

Proof of Theorem dfac11
Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfac3 9536 . . 3 (CHOICE ↔ ∀𝑎𝑐𝑑𝑎 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑))
2 raleq 3361 . . . . . 6 (𝑎 = 𝑥 → (∀𝑑𝑎 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) ↔ ∀𝑑𝑥 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑)))
32exbidv 1922 . . . . 5 (𝑎 = 𝑥 → (∃𝑐𝑑𝑎 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) ↔ ∃𝑐𝑑𝑥 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑)))
43cbvalvw 2043 . . . 4 (∀𝑎𝑐𝑑𝑎 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) ↔ ∀𝑥𝑐𝑑𝑥 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑))
5 neeq1 3052 . . . . . . . . . 10 (𝑑 = 𝑧 → (𝑑 ≠ ∅ ↔ 𝑧 ≠ ∅))
6 fveq2 6649 . . . . . . . . . . 11 (𝑑 = 𝑧 → (𝑐𝑑) = (𝑐𝑧))
7 id 22 . . . . . . . . . . 11 (𝑑 = 𝑧𝑑 = 𝑧)
86, 7eleq12d 2887 . . . . . . . . . 10 (𝑑 = 𝑧 → ((𝑐𝑑) ∈ 𝑑 ↔ (𝑐𝑧) ∈ 𝑧))
95, 8imbi12d 348 . . . . . . . . 9 (𝑑 = 𝑧 → ((𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) ↔ (𝑧 ≠ ∅ → (𝑐𝑧) ∈ 𝑧)))
109cbvralvw 3399 . . . . . . . 8 (∀𝑑𝑥 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) ↔ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑐𝑧) ∈ 𝑧))
11 fveq2 6649 . . . . . . . . . . . . . . 15 (𝑏 = 𝑧 → (𝑐𝑏) = (𝑐𝑧))
1211sneqd 4540 . . . . . . . . . . . . . 14 (𝑏 = 𝑧 → {(𝑐𝑏)} = {(𝑐𝑧)})
13 eqid 2801 . . . . . . . . . . . . . 14 (𝑏𝑥 ↦ {(𝑐𝑏)}) = (𝑏𝑥 ↦ {(𝑐𝑏)})
14 snex 5300 . . . . . . . . . . . . . 14 {(𝑐𝑧)} ∈ V
1512, 13, 14fvmpt 6749 . . . . . . . . . . . . 13 (𝑧𝑥 → ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) = {(𝑐𝑧)})
16153ad2ant1 1130 . . . . . . . . . . . 12 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) = {(𝑐𝑧)})
17 simp3 1135 . . . . . . . . . . . . . . . 16 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → (𝑐𝑧) ∈ 𝑧)
1817snssd 4705 . . . . . . . . . . . . . . 15 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → {(𝑐𝑧)} ⊆ 𝑧)
1914elpw 4504 . . . . . . . . . . . . . . 15 ({(𝑐𝑧)} ∈ 𝒫 𝑧 ↔ {(𝑐𝑧)} ⊆ 𝑧)
2018, 19sylibr 237 . . . . . . . . . . . . . 14 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → {(𝑐𝑧)} ∈ 𝒫 𝑧)
21 snfi 8581 . . . . . . . . . . . . . . 15 {(𝑐𝑧)} ∈ Fin
2221a1i 11 . . . . . . . . . . . . . 14 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → {(𝑐𝑧)} ∈ Fin)
2320, 22elind 4124 . . . . . . . . . . . . 13 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → {(𝑐𝑧)} ∈ (𝒫 𝑧 ∩ Fin))
24 fvex 6662 . . . . . . . . . . . . . . 15 (𝑐𝑧) ∈ V
2524snnz 4675 . . . . . . . . . . . . . 14 {(𝑐𝑧)} ≠ ∅
2625a1i 11 . . . . . . . . . . . . 13 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → {(𝑐𝑧)} ≠ ∅)
27 eldifsn 4683 . . . . . . . . . . . . 13 ({(𝑐𝑧)} ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}) ↔ ({(𝑐𝑧)} ∈ (𝒫 𝑧 ∩ Fin) ∧ {(𝑐𝑧)} ≠ ∅))
2823, 26, 27sylanbrc 586 . . . . . . . . . . . 12 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → {(𝑐𝑧)} ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))
2916, 28eqeltrd 2893 . . . . . . . . . . 11 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))
30293exp 1116 . . . . . . . . . 10 (𝑧𝑥 → (𝑧 ≠ ∅ → ((𝑐𝑧) ∈ 𝑧 → ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))))
3130a2d 29 . . . . . . . . 9 (𝑧𝑥 → ((𝑧 ≠ ∅ → (𝑐𝑧) ∈ 𝑧) → (𝑧 ≠ ∅ → ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))))
3231ralimia 3129 . . . . . . . 8 (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑐𝑧) ∈ 𝑧) → ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
3310, 32sylbi 220 . . . . . . 7 (∀𝑑𝑥 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) → ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
34 vex 3447 . . . . . . . . 9 𝑥 ∈ V
3534mptex 6967 . . . . . . . 8 (𝑏𝑥 ↦ {(𝑐𝑏)}) ∈ V
36 fveq1 6648 . . . . . . . . . . 11 (𝑓 = (𝑏𝑥 ↦ {(𝑐𝑏)}) → (𝑓𝑧) = ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧))
3736eleq1d 2877 . . . . . . . . . 10 (𝑓 = (𝑏𝑥 ↦ {(𝑐𝑏)}) → ((𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}) ↔ ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
3837imbi2d 344 . . . . . . . . 9 (𝑓 = (𝑏𝑥 ↦ {(𝑐𝑏)}) → ((𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) ↔ (𝑧 ≠ ∅ → ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))))
3938ralbidv 3165 . . . . . . . 8 (𝑓 = (𝑏𝑥 ↦ {(𝑐𝑏)}) → (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) ↔ ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))))
4035, 39spcev 3558 . . . . . . 7 (∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
4133, 40syl 17 . . . . . 6 (∀𝑑𝑥 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) → ∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
4241exlimiv 1931 . . . . 5 (∃𝑐𝑑𝑥 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) → ∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
4342alimi 1813 . . . 4 (∀𝑥𝑐𝑑𝑥 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) → ∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
444, 43sylbi 220 . . 3 (∀𝑎𝑐𝑑𝑎 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) → ∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
451, 44sylbi 220 . 2 (CHOICE → ∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
46 fvex 6662 . . . . . . 7 (𝑅1‘(rank‘𝑎)) ∈ V
4746pwex 5249 . . . . . 6 𝒫 (𝑅1‘(rank‘𝑎)) ∈ V
48 raleq 3361 . . . . . . 7 (𝑥 = 𝒫 (𝑅1‘(rank‘𝑎)) → (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) ↔ ∀𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))))
4948exbidv 1922 . . . . . 6 (𝑥 = 𝒫 (𝑅1‘(rank‘𝑎)) → (∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) ↔ ∃𝑓𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))))
5047, 49spcv 3557 . . . . 5 (∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∃𝑓𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
51 rankon 9212 . . . . . . . 8 (rank‘𝑎) ∈ On
5251a1i 11 . . . . . . 7 (∀𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → (rank‘𝑎) ∈ On)
53 id 22 . . . . . . 7 (∀𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∀𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
5452, 53aomclem8 39998 . . . . . 6 (∀𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∃𝑏 𝑏 We (𝑅1‘(rank‘𝑎)))
5554exlimiv 1931 . . . . 5 (∃𝑓𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∃𝑏 𝑏 We (𝑅1‘(rank‘𝑎)))
56 vex 3447 . . . . . 6 𝑎 ∈ V
57 r1rankid 9276 . . . . . 6 (𝑎 ∈ V → 𝑎 ⊆ (𝑅1‘(rank‘𝑎)))
58 wess 5510 . . . . . . 7 (𝑎 ⊆ (𝑅1‘(rank‘𝑎)) → (𝑏 We (𝑅1‘(rank‘𝑎)) → 𝑏 We 𝑎))
5958eximdv 1918 . . . . . 6 (𝑎 ⊆ (𝑅1‘(rank‘𝑎)) → (∃𝑏 𝑏 We (𝑅1‘(rank‘𝑎)) → ∃𝑏 𝑏 We 𝑎))
6056, 57, 59mp2b 10 . . . . 5 (∃𝑏 𝑏 We (𝑅1‘(rank‘𝑎)) → ∃𝑏 𝑏 We 𝑎)
6150, 55, 603syl 18 . . . 4 (∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∃𝑏 𝑏 We 𝑎)
6261alrimiv 1928 . . 3 (∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∀𝑎𝑏 𝑏 We 𝑎)
63 dfac8 9550 . . 3 (CHOICE ↔ ∀𝑎𝑏 𝑏 We 𝑎)
6462, 63sylibr 237 . 2 (∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → CHOICE)
6545, 64impbii 212 1 (CHOICE ↔ ∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  w3a 1084  wal 1536   = wceq 1538  wex 1781  wcel 2112  wne 2990  wral 3109  Vcvv 3444  cdif 3881  cin 3883  wss 3884  c0 4246  𝒫 cpw 4500  {csn 4528  cmpt 5113   We wwe 5481  Oncon0 6163  cfv 6328  Fincfn 8496  𝑅1cr1 9179  rankcrnk 9180  CHOICEwac 9530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-reg 9044  ax-inf2 9092
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-er 8276  df-map 8395  df-en 8497  df-fin 8500  df-sup 8894  df-r1 9181  df-rank 9182  df-card 9356  df-ac 9531
This theorem is referenced by: (None)
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