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Theorem dfac3 9257
Description: Equivalence of two versions of the Axiom of Choice. The left-hand side is defined as the Axiom of Choice (first form) of [Enderton] p. 49. The right-hand side is the Axiom of Choice of [TakeutiZaring] p. 83. The proof does not depend on AC. (Contributed by NM, 24-Mar-2004.) (Revised by Stefan O'Rear, 22-Feb-2015.)
Assertion
Ref Expression
dfac3 (CHOICE ↔ ∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
Distinct variable group:   𝑥,𝑓,𝑧

Proof of Theorem dfac3
Dummy variables 𝑦 𝑤 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ac 9252 . 2 (CHOICE ↔ ∀𝑦𝑓(𝑓𝑦𝑓 Fn dom 𝑦))
2 vex 3417 . . . . . . . 8 𝑥 ∈ V
3 vuniex 7214 . . . . . . . 8 𝑥 ∈ V
42, 3xpex 7223 . . . . . . 7 (𝑥 × 𝑥) ∈ V
5 simpl 476 . . . . . . . . . 10 ((𝑤𝑥𝑣𝑤) → 𝑤𝑥)
6 elunii 4663 . . . . . . . . . . 11 ((𝑣𝑤𝑤𝑥) → 𝑣 𝑥)
76ancoms 452 . . . . . . . . . 10 ((𝑤𝑥𝑣𝑤) → 𝑣 𝑥)
85, 7jca 507 . . . . . . . . 9 ((𝑤𝑥𝑣𝑤) → (𝑤𝑥𝑣 𝑥))
98ssopab2i 5229 . . . . . . . 8 {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣 𝑥)}
10 df-xp 5348 . . . . . . . 8 (𝑥 × 𝑥) = {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣 𝑥)}
119, 10sseqtr4i 3863 . . . . . . 7 {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ⊆ (𝑥 × 𝑥)
124, 11ssexi 5028 . . . . . 6 {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ∈ V
13 sseq2 3852 . . . . . . . 8 (𝑦 = {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} → (𝑓𝑦𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}))
14 dmeq 5556 . . . . . . . . 9 (𝑦 = {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} → dom 𝑦 = dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)})
1514fneq2d 6215 . . . . . . . 8 (𝑦 = {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} → (𝑓 Fn dom 𝑦𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}))
1613, 15anbi12d 624 . . . . . . 7 (𝑦 = {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} → ((𝑓𝑦𝑓 Fn dom 𝑦) ↔ (𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)})))
1716exbidv 2020 . . . . . 6 (𝑦 = {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} → (∃𝑓(𝑓𝑦𝑓 Fn dom 𝑦) ↔ ∃𝑓(𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)})))
1812, 17spcv 3516 . . . . 5 (∀𝑦𝑓(𝑓𝑦𝑓 Fn dom 𝑦) → ∃𝑓(𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}))
19 fndm 6223 . . . . . . . . . . . . 13 (𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} → dom 𝑓 = dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)})
20 eleq2 2895 . . . . . . . . . . . . . 14 (dom 𝑓 = dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} → (𝑧 ∈ dom 𝑓𝑧 ∈ dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}))
21 dmopab 5567 . . . . . . . . . . . . . . . 16 dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} = {𝑤 ∣ ∃𝑣(𝑤𝑥𝑣𝑤)}
2221eleq2i 2898 . . . . . . . . . . . . . . 15 (𝑧 ∈ dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ↔ 𝑧 ∈ {𝑤 ∣ ∃𝑣(𝑤𝑥𝑣𝑤)})
23 vex 3417 . . . . . . . . . . . . . . . 16 𝑧 ∈ V
24 elequ1 2171 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑧 → (𝑤𝑥𝑧𝑥))
25 eleq2 2895 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑧 → (𝑣𝑤𝑣𝑧))
2624, 25anbi12d 624 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑧 → ((𝑤𝑥𝑣𝑤) ↔ (𝑧𝑥𝑣𝑧)))
2726exbidv 2020 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑧 → (∃𝑣(𝑤𝑥𝑣𝑤) ↔ ∃𝑣(𝑧𝑥𝑣𝑧)))
2823, 27elab 3571 . . . . . . . . . . . . . . 15 (𝑧 ∈ {𝑤 ∣ ∃𝑣(𝑤𝑥𝑣𝑤)} ↔ ∃𝑣(𝑧𝑥𝑣𝑧))
29 19.42v 2052 . . . . . . . . . . . . . . . 16 (∃𝑣(𝑧𝑥𝑣𝑧) ↔ (𝑧𝑥 ∧ ∃𝑣 𝑣𝑧))
30 n0 4160 . . . . . . . . . . . . . . . . 17 (𝑧 ≠ ∅ ↔ ∃𝑣 𝑣𝑧)
3130anbi2i 616 . . . . . . . . . . . . . . . 16 ((𝑧𝑥𝑧 ≠ ∅) ↔ (𝑧𝑥 ∧ ∃𝑣 𝑣𝑧))
3229, 31bitr4i 270 . . . . . . . . . . . . . . 15 (∃𝑣(𝑧𝑥𝑣𝑧) ↔ (𝑧𝑥𝑧 ≠ ∅))
3322, 28, 323bitrri 290 . . . . . . . . . . . . . 14 ((𝑧𝑥𝑧 ≠ ∅) ↔ 𝑧 ∈ dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)})
3420, 33syl6rbbr 282 . . . . . . . . . . . . 13 (dom 𝑓 = dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} → ((𝑧𝑥𝑧 ≠ ∅) ↔ 𝑧 ∈ dom 𝑓))
3519, 34syl 17 . . . . . . . . . . . 12 (𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} → ((𝑧𝑥𝑧 ≠ ∅) ↔ 𝑧 ∈ dom 𝑓))
3635adantl 475 . . . . . . . . . . 11 ((𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}) → ((𝑧𝑥𝑧 ≠ ∅) ↔ 𝑧 ∈ dom 𝑓))
37 fnfun 6221 . . . . . . . . . . . 12 (𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} → Fun 𝑓)
38 funfvima3 6751 . . . . . . . . . . . . 13 ((Fun 𝑓𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}) → (𝑧 ∈ dom 𝑓 → (𝑓𝑧) ∈ ({⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} “ {𝑧})))
3938ancoms 452 . . . . . . . . . . . 12 ((𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ∧ Fun 𝑓) → (𝑧 ∈ dom 𝑓 → (𝑓𝑧) ∈ ({⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} “ {𝑧})))
4037, 39sylan2 586 . . . . . . . . . . 11 ((𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}) → (𝑧 ∈ dom 𝑓 → (𝑓𝑧) ∈ ({⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} “ {𝑧})))
4136, 40sylbid 232 . . . . . . . . . 10 ((𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}) → ((𝑧𝑥𝑧 ≠ ∅) → (𝑓𝑧) ∈ ({⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} “ {𝑧})))
4241imp 397 . . . . . . . . 9 (((𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}) ∧ (𝑧𝑥𝑧 ≠ ∅)) → (𝑓𝑧) ∈ ({⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} “ {𝑧}))
43 ibar 524 . . . . . . . . . . . . 13 (𝑧𝑥 → (𝑢𝑧 ↔ (𝑧𝑥𝑢𝑧)))
4443abbi2dv 2947 . . . . . . . . . . . 12 (𝑧𝑥𝑧 = {𝑢 ∣ (𝑧𝑥𝑢𝑧)})
45 imasng 5728 . . . . . . . . . . . . . 14 (𝑧 ∈ V → ({⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} “ {𝑧}) = {𝑢𝑧{⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}𝑢})
4623, 45ax-mp 5 . . . . . . . . . . . . 13 ({⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} “ {𝑧}) = {𝑢𝑧{⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}𝑢}
47 vex 3417 . . . . . . . . . . . . . . 15 𝑢 ∈ V
48 elequ1 2171 . . . . . . . . . . . . . . . 16 (𝑣 = 𝑢 → (𝑣𝑧𝑢𝑧))
4948anbi2d 622 . . . . . . . . . . . . . . 15 (𝑣 = 𝑢 → ((𝑧𝑥𝑣𝑧) ↔ (𝑧𝑥𝑢𝑧)))
50 eqid 2825 . . . . . . . . . . . . . . 15 {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} = {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}
5123, 47, 26, 49, 50brab 5224 . . . . . . . . . . . . . 14 (𝑧{⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}𝑢 ↔ (𝑧𝑥𝑢𝑧))
5251abbii 2944 . . . . . . . . . . . . 13 {𝑢𝑧{⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}𝑢} = {𝑢 ∣ (𝑧𝑥𝑢𝑧)}
5346, 52eqtri 2849 . . . . . . . . . . . 12 ({⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} “ {𝑧}) = {𝑢 ∣ (𝑧𝑥𝑢𝑧)}
5444, 53syl6reqr 2880 . . . . . . . . . . 11 (𝑧𝑥 → ({⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} “ {𝑧}) = 𝑧)
5554eleq2d 2892 . . . . . . . . . 10 (𝑧𝑥 → ((𝑓𝑧) ∈ ({⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} “ {𝑧}) ↔ (𝑓𝑧) ∈ 𝑧))
5655ad2antrl 719 . . . . . . . . 9 (((𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}) ∧ (𝑧𝑥𝑧 ≠ ∅)) → ((𝑓𝑧) ∈ ({⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} “ {𝑧}) ↔ (𝑓𝑧) ∈ 𝑧))
5742, 56mpbid 224 . . . . . . . 8 (((𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}) ∧ (𝑧𝑥𝑧 ≠ ∅)) → (𝑓𝑧) ∈ 𝑧)
5857exp32 413 . . . . . . 7 ((𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}) → (𝑧𝑥 → (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
5958ralrimiv 3174 . . . . . 6 ((𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}) → ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
6059eximi 1933 . . . . 5 (∃𝑓(𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}) → ∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
6118, 60syl 17 . . . 4 (∀𝑦𝑓(𝑓𝑦𝑓 Fn dom 𝑦) → ∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
6261alrimiv 2026 . . 3 (∀𝑦𝑓(𝑓𝑦𝑓 Fn dom 𝑦) → ∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
63 eqid 2825 . . . . 5 (𝑤 ∈ dom 𝑦 ↦ (𝑓‘{𝑢𝑤𝑦𝑢})) = (𝑤 ∈ dom 𝑦 ↦ (𝑓‘{𝑢𝑤𝑦𝑢}))
6463aceq3lem 9256 . . . 4 (∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∃𝑓(𝑓𝑦𝑓 Fn dom 𝑦))
6564alrimiv 2026 . . 3 (∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∀𝑦𝑓(𝑓𝑦𝑓 Fn dom 𝑦))
6662, 65impbii 201 . 2 (∀𝑦𝑓(𝑓𝑦𝑓 Fn dom 𝑦) ↔ ∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
671, 66bitri 267 1 (CHOICE ↔ ∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  wal 1654   = wceq 1656  wex 1878  wcel 2164  {cab 2811  wne 2999  wral 3117  Vcvv 3414  wss 3798  c0 4144  {csn 4397   cuni 4658   class class class wbr 4873  {copab 4935  cmpt 4952   × cxp 5340  dom cdm 5342  cima 5345  Fun wfun 6117   Fn wfn 6118  cfv 6123  CHOICEwac 9251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-fv 6131  df-ac 9252
This theorem is referenced by:  dfac4  9258  dfac5  9264  dfac2a  9265  dfac2b  9266  dfac2OLD  9268  dfac8  9272  dfac9  9273  ac4  9612  dfac11  38468
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