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Theorem dfac11 38158
Description: The right-hand side of this theorem (compare with ac4 9499), sometimes known as the "axiom of multiple choice", is a choice equivalent. Curiously, this statement cannot be proved without ax-reg 8653, 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 9144 . . 3 (CHOICE ↔ ∀𝑎𝑐𝑑𝑎 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑))
2 raleq 3287 . . . . . 6 (𝑎 = 𝑥 → (∀𝑑𝑎 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) ↔ ∀𝑑𝑥 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑)))
32exbidv 2002 . . . . 5 (𝑎 = 𝑥 → (∃𝑐𝑑𝑎 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) ↔ ∃𝑐𝑑𝑥 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑)))
43cbvalvw 2125 . . . 4 (∀𝑎𝑐𝑑𝑎 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) ↔ ∀𝑥𝑐𝑑𝑥 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑))
5 neeq1 3005 . . . . . . . . . 10 (𝑑 = 𝑧 → (𝑑 ≠ ∅ ↔ 𝑧 ≠ ∅))
6 fveq2 6332 . . . . . . . . . . 11 (𝑑 = 𝑧 → (𝑐𝑑) = (𝑐𝑧))
7 id 22 . . . . . . . . . . 11 (𝑑 = 𝑧𝑑 = 𝑧)
86, 7eleq12d 2844 . . . . . . . . . 10 (𝑑 = 𝑧 → ((𝑐𝑑) ∈ 𝑑 ↔ (𝑐𝑧) ∈ 𝑧))
95, 8imbi12d 333 . . . . . . . . 9 (𝑑 = 𝑧 → ((𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) ↔ (𝑧 ≠ ∅ → (𝑐𝑧) ∈ 𝑧)))
109cbvralv 3320 . . . . . . . 8 (∀𝑑𝑥 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) ↔ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑐𝑧) ∈ 𝑧))
11 fveq2 6332 . . . . . . . . . . . . . . 15 (𝑏 = 𝑧 → (𝑐𝑏) = (𝑐𝑧))
1211sneqd 4328 . . . . . . . . . . . . . 14 (𝑏 = 𝑧 → {(𝑐𝑏)} = {(𝑐𝑧)})
13 eqid 2771 . . . . . . . . . . . . . 14 (𝑏𝑥 ↦ {(𝑐𝑏)}) = (𝑏𝑥 ↦ {(𝑐𝑏)})
14 snex 5036 . . . . . . . . . . . . . 14 {(𝑐𝑧)} ∈ V
1512, 13, 14fvmpt 6424 . . . . . . . . . . . . 13 (𝑧𝑥 → ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) = {(𝑐𝑧)})
16153ad2ant1 1127 . . . . . . . . . . . 12 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) = {(𝑐𝑧)})
17 simp3 1132 . . . . . . . . . . . . . . . 16 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → (𝑐𝑧) ∈ 𝑧)
1817snssd 4475 . . . . . . . . . . . . . . 15 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → {(𝑐𝑧)} ⊆ 𝑧)
1914elpw 4303 . . . . . . . . . . . . . . 15 ({(𝑐𝑧)} ∈ 𝒫 𝑧 ↔ {(𝑐𝑧)} ⊆ 𝑧)
2018, 19sylibr 224 . . . . . . . . . . . . . 14 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → {(𝑐𝑧)} ∈ 𝒫 𝑧)
21 snfi 8194 . . . . . . . . . . . . . . 15 {(𝑐𝑧)} ∈ Fin
2221a1i 11 . . . . . . . . . . . . . 14 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → {(𝑐𝑧)} ∈ Fin)
2320, 22elind 3949 . . . . . . . . . . . . 13 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → {(𝑐𝑧)} ∈ (𝒫 𝑧 ∩ Fin))
24 fvex 6342 . . . . . . . . . . . . . . 15 (𝑐𝑧) ∈ V
2524snnz 4444 . . . . . . . . . . . . . 14 {(𝑐𝑧)} ≠ ∅
2625a1i 11 . . . . . . . . . . . . 13 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → {(𝑐𝑧)} ≠ ∅)
27 eldifsn 4453 . . . . . . . . . . . . 13 ({(𝑐𝑧)} ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}) ↔ ({(𝑐𝑧)} ∈ (𝒫 𝑧 ∩ Fin) ∧ {(𝑐𝑧)} ≠ ∅))
2823, 26, 27sylanbrc 572 . . . . . . . . . . . 12 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → {(𝑐𝑧)} ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))
2916, 28eqeltrd 2850 . . . . . . . . . . 11 ((𝑧𝑥𝑧 ≠ ∅ ∧ (𝑐𝑧) ∈ 𝑧) → ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))
30293exp 1112 . . . . . . . . . 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 3099 . . . . . . . 8 (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑐𝑧) ∈ 𝑧) → ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
3310, 32sylbi 207 . . . . . . 7 (∀𝑑𝑥 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) → ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
34 vex 3354 . . . . . . . . 9 𝑥 ∈ V
3534mptex 6630 . . . . . . . 8 (𝑏𝑥 ↦ {(𝑐𝑏)}) ∈ V
36 fveq1 6331 . . . . . . . . . . 11 (𝑓 = (𝑏𝑥 ↦ {(𝑐𝑏)}) → (𝑓𝑧) = ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧))
3736eleq1d 2835 . . . . . . . . . 10 (𝑓 = (𝑏𝑥 ↦ {(𝑐𝑏)}) → ((𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}) ↔ ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
3837imbi2d 329 . . . . . . . . 9 (𝑓 = (𝑏𝑥 ↦ {(𝑐𝑏)}) → ((𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) ↔ (𝑧 ≠ ∅ → ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))))
3938ralbidv 3135 . . . . . . . 8 (𝑓 = (𝑏𝑥 ↦ {(𝑐𝑏)}) → (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) ↔ ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑏𝑥 ↦ {(𝑐𝑏)})‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))))
4035, 39spcev 3451 . . . . . . 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 2010 . . . . 5 (∃𝑐𝑑𝑥 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) → ∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
4342alimi 1887 . . . 4 (∀𝑥𝑐𝑑𝑥 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) → ∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
444, 43sylbi 207 . . 3 (∀𝑎𝑐𝑑𝑎 (𝑑 ≠ ∅ → (𝑐𝑑) ∈ 𝑑) → ∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
451, 44sylbi 207 . 2 (CHOICE → ∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
46 fvex 6342 . . . . . . 7 (𝑅1‘(rank‘𝑎)) ∈ V
4746pwex 4981 . . . . . 6 𝒫 (𝑅1‘(rank‘𝑎)) ∈ V
48 raleq 3287 . . . . . . 7 (𝑥 = 𝒫 (𝑅1‘(rank‘𝑎)) → (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) ↔ ∀𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))))
4948exbidv 2002 . . . . . 6 (𝑥 = 𝒫 (𝑅1‘(rank‘𝑎)) → (∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) ↔ ∃𝑓𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅}))))
5047, 49spcv 3450 . . . . 5 (∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∃𝑓𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
51 rankon 8822 . . . . . . . 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 38157 . . . . . 6 (∀𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∃𝑏 𝑏 We (𝑅1‘(rank‘𝑎)))
5554exlimiv 2010 . . . . 5 (∃𝑓𝑧 ∈ 𝒫 (𝑅1‘(rank‘𝑎))(𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∃𝑏 𝑏 We (𝑅1‘(rank‘𝑎)))
56 vex 3354 . . . . . 6 𝑎 ∈ V
57 r1rankid 8886 . . . . . 6 (𝑎 ∈ V → 𝑎 ⊆ (𝑅1‘(rank‘𝑎)))
58 wess 5236 . . . . . . 7 (𝑎 ⊆ (𝑅1‘(rank‘𝑎)) → (𝑏 We (𝑅1‘(rank‘𝑎)) → 𝑏 We 𝑎))
5958eximdv 1998 . . . . . 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 2007 . . 3 (∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → ∀𝑎𝑏 𝑏 We 𝑎)
63 dfac8 9159 . . 3 (CHOICE ↔ ∀𝑎𝑏 𝑏 We 𝑎)
6462, 63sylibr 224 . 2 (∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})) → CHOICE)
6545, 64impbii 199 1 (CHOICE ↔ ∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1071  wal 1629   = wceq 1631  wex 1852  wcel 2145  wne 2943  wral 3061  Vcvv 3351  cdif 3720  cin 3722  wss 3723  c0 4063  𝒫 cpw 4297  {csn 4316  cmpt 4863   We wwe 5207  Oncon0 5866  cfv 6031  Fincfn 8109  𝑅1cr1 8789  rankcrnk 8790  CHOICEwac 9138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-reg 8653  ax-inf2 8702
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-er 7896  df-map 8011  df-en 8110  df-fin 8113  df-sup 8504  df-r1 8791  df-rank 8792  df-card 8965  df-ac 9139
This theorem is referenced by: (None)
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