MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfac3 Structured version   Visualization version   GIF version

Theorem dfac3 8941
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 8936 . 2 (CHOICE ↔ ∀𝑦𝑓(𝑓𝑦𝑓 Fn dom 𝑦))
2 vex 3201 . . . . . . . 8 𝑥 ∈ V
3 vuniex 6951 . . . . . . . 8 𝑥 ∈ V
42, 3xpex 6959 . . . . . . 7 (𝑥 × 𝑥) ∈ V
5 simpl 473 . . . . . . . . . 10 ((𝑤𝑥𝑣𝑤) → 𝑤𝑥)
6 elunii 4439 . . . . . . . . . . 11 ((𝑣𝑤𝑤𝑥) → 𝑣 𝑥)
76ancoms 469 . . . . . . . . . 10 ((𝑤𝑥𝑣𝑤) → 𝑣 𝑥)
85, 7jca 554 . . . . . . . . 9 ((𝑤𝑥𝑣𝑤) → (𝑤𝑥𝑣 𝑥))
98ssopab2i 5001 . . . . . . . 8 {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣 𝑥)}
10 df-xp 5118 . . . . . . . 8 (𝑥 × 𝑥) = {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣 𝑥)}
119, 10sseqtr4i 3636 . . . . . . 7 {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ⊆ (𝑥 × 𝑥)
124, 11ssexi 4801 . . . . . 6 {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ∈ V
13 sseq2 3625 . . . . . . . 8 (𝑦 = {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} → (𝑓𝑦𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}))
14 dmeq 5322 . . . . . . . . 9 (𝑦 = {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} → dom 𝑦 = dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)})
1514fneq2d 5980 . . . . . . . 8 (𝑦 = {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} → (𝑓 Fn dom 𝑦𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}))
1613, 15anbi12d 747 . . . . . . 7 (𝑦 = {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} → ((𝑓𝑦𝑓 Fn dom 𝑦) ↔ (𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)})))
1716exbidv 1849 . . . . . 6 (𝑦 = {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} → (∃𝑓(𝑓𝑦𝑓 Fn dom 𝑦) ↔ ∃𝑓(𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)})))
1812, 17spcv 3297 . . . . 5 (∀𝑦𝑓(𝑓𝑦𝑓 Fn dom 𝑦) → ∃𝑓(𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}))
19 fndm 5988 . . . . . . . . . . . . 13 (𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} → dom 𝑓 = dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)})
20 eleq2 2689 . . . . . . . . . . . . . 14 (dom 𝑓 = dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} → (𝑧 ∈ dom 𝑓𝑧 ∈ dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}))
21 dmopab 5333 . . . . . . . . . . . . . . . 16 dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} = {𝑤 ∣ ∃𝑣(𝑤𝑥𝑣𝑤)}
2221eleq2i 2692 . . . . . . . . . . . . . . 15 (𝑧 ∈ dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ↔ 𝑧 ∈ {𝑤 ∣ ∃𝑣(𝑤𝑥𝑣𝑤)})
23 vex 3201 . . . . . . . . . . . . . . . 16 𝑧 ∈ V
24 elequ1 1996 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑧 → (𝑤𝑥𝑧𝑥))
25 eleq2 2689 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑧 → (𝑣𝑤𝑣𝑧))
2624, 25anbi12d 747 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑧 → ((𝑤𝑥𝑣𝑤) ↔ (𝑧𝑥𝑣𝑧)))
2726exbidv 1849 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑧 → (∃𝑣(𝑤𝑥𝑣𝑤) ↔ ∃𝑣(𝑧𝑥𝑣𝑧)))
2823, 27elab 3348 . . . . . . . . . . . . . . 15 (𝑧 ∈ {𝑤 ∣ ∃𝑣(𝑤𝑥𝑣𝑤)} ↔ ∃𝑣(𝑧𝑥𝑣𝑧))
29 19.42v 1917 . . . . . . . . . . . . . . . 16 (∃𝑣(𝑧𝑥𝑣𝑧) ↔ (𝑧𝑥 ∧ ∃𝑣 𝑣𝑧))
30 n0 3929 . . . . . . . . . . . . . . . . 17 (𝑧 ≠ ∅ ↔ ∃𝑣 𝑣𝑧)
3130anbi2i 730 . . . . . . . . . . . . . . . 16 ((𝑧𝑥𝑧 ≠ ∅) ↔ (𝑧𝑥 ∧ ∃𝑣 𝑣𝑧))
3229, 31bitr4i 267 . . . . . . . . . . . . . . 15 (∃𝑣(𝑧𝑥𝑣𝑧) ↔ (𝑧𝑥𝑧 ≠ ∅))
3322, 28, 323bitrri 287 . . . . . . . . . . . . . 14 ((𝑧𝑥𝑧 ≠ ∅) ↔ 𝑧 ∈ dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)})
3420, 33syl6rbbr 279 . . . . . . . . . . . . 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 482 . . . . . . . . . . 11 ((𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}) → ((𝑧𝑥𝑧 ≠ ∅) ↔ 𝑧 ∈ dom 𝑓))
37 fnfun 5986 . . . . . . . . . . . 12 (𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} → Fun 𝑓)
38 funfvima3 6492 . . . . . . . . . . . . 13 ((Fun 𝑓𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}) → (𝑧 ∈ dom 𝑓 → (𝑓𝑧) ∈ ({⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} “ {𝑧})))
3938ancoms 469 . . . . . . . . . . . 12 ((𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ∧ Fun 𝑓) → (𝑧 ∈ dom 𝑓 → (𝑓𝑧) ∈ ({⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} “ {𝑧})))
4037, 39sylan2 491 . . . . . . . . . . 11 ((𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}) → (𝑧 ∈ dom 𝑓 → (𝑓𝑧) ∈ ({⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} “ {𝑧})))
4136, 40sylbid 230 . . . . . . . . . 10 ((𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}) → ((𝑧𝑥𝑧 ≠ ∅) → (𝑓𝑧) ∈ ({⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} “ {𝑧})))
4241imp 445 . . . . . . . . 9 (((𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}) ∧ (𝑧𝑥𝑧 ≠ ∅)) → (𝑓𝑧) ∈ ({⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} “ {𝑧}))
43 ibar 525 . . . . . . . . . . . . 13 (𝑧𝑥 → (𝑢𝑧 ↔ (𝑧𝑥𝑢𝑧)))
4443abbi2dv 2741 . . . . . . . . . . . 12 (𝑧𝑥𝑧 = {𝑢 ∣ (𝑧𝑥𝑢𝑧)})
45 imasng 5485 . . . . . . . . . . . . . 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 3201 . . . . . . . . . . . . . . 15 𝑢 ∈ V
48 elequ1 1996 . . . . . . . . . . . . . . . 16 (𝑣 = 𝑢 → (𝑣𝑧𝑢𝑧))
4948anbi2d 740 . . . . . . . . . . . . . . 15 (𝑣 = 𝑢 → ((𝑧𝑥𝑣𝑧) ↔ (𝑧𝑥𝑢𝑧)))
50 eqid 2621 . . . . . . . . . . . . . . 15 {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} = {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}
5123, 47, 26, 49, 50brab 4996 . . . . . . . . . . . . . 14 (𝑧{⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}𝑢 ↔ (𝑧𝑥𝑢𝑧))
5251abbii 2738 . . . . . . . . . . . . 13 {𝑢𝑧{⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}𝑢} = {𝑢 ∣ (𝑧𝑥𝑢𝑧)}
5346, 52eqtri 2643 . . . . . . . . . . . 12 ({⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} “ {𝑧}) = {𝑢 ∣ (𝑧𝑥𝑢𝑧)}
5444, 53syl6reqr 2674 . . . . . . . . . . 11 (𝑧𝑥 → ({⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} “ {𝑧}) = 𝑧)
5554eleq2d 2686 . . . . . . . . . 10 (𝑧𝑥 → ((𝑓𝑧) ∈ ({⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} “ {𝑧}) ↔ (𝑓𝑧) ∈ 𝑧))
5655ad2antrl 764 . . . . . . . . 9 (((𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}) ∧ (𝑧𝑥𝑧 ≠ ∅)) → ((𝑓𝑧) ∈ ({⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} “ {𝑧}) ↔ (𝑓𝑧) ∈ 𝑧))
5742, 56mpbid 222 . . . . . . . 8 (((𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}) ∧ (𝑧𝑥𝑧 ≠ ∅)) → (𝑓𝑧) ∈ 𝑧)
5857exp32 631 . . . . . . 7 ((𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}) → (𝑧𝑥 → (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
5958ralrimiv 2964 . . . . . 6 ((𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}) → ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
6059eximi 1761 . . . . 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 1854 . . 3 (∀𝑦𝑓(𝑓𝑦𝑓 Fn dom 𝑦) → ∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
63 eqid 2621 . . . . 5 (𝑤 ∈ dom 𝑦 ↦ (𝑓‘{𝑢𝑤𝑦𝑢})) = (𝑤 ∈ dom 𝑦 ↦ (𝑓‘{𝑢𝑤𝑦𝑢}))
6463aceq3lem 8940 . . . 4 (∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∃𝑓(𝑓𝑦𝑓 Fn dom 𝑦))
6564alrimiv 1854 . . 3 (∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∀𝑦𝑓(𝑓𝑦𝑓 Fn dom 𝑦))
6662, 65impbii 199 . 2 (∀𝑦𝑓(𝑓𝑦𝑓 Fn dom 𝑦) ↔ ∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
671, 66bitri 264 1 (CHOICE ↔ ∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1480   = wceq 1482  wex 1703  wcel 1989  {cab 2607  wne 2793  wral 2911  Vcvv 3198  wss 3572  c0 3913  {csn 4175   cuni 4434   class class class wbr 4651  {copab 4710  cmpt 4727   × cxp 5110  dom cdm 5112  cima 5115  Fun wfun 5880   Fn wfn 5881  cfv 5886  CHOICEwac 8935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-fv 5894  df-ac 8936
This theorem is referenced by:  dfac4  8942  dfac5  8948  dfac2a  8949  dfac2  8950  dfac8  8954  dfac9  8955  ac4  9294  dfac11  37458
  Copyright terms: Public domain W3C validator