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Theorem dfac3 8895
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 8890 . 2 (CHOICE ↔ ∀𝑦𝑓(𝑓𝑦𝑓 Fn dom 𝑦))
2 vex 3192 . . . . . . . 8 𝑥 ∈ V
3 vuniex 6914 . . . . . . . 8 𝑥 ∈ V
42, 3xpex 6922 . . . . . . 7 (𝑥 × 𝑥) ∈ V
5 simpl 473 . . . . . . . . . 10 ((𝑤𝑥𝑣𝑤) → 𝑤𝑥)
6 elunii 4412 . . . . . . . . . . 11 ((𝑣𝑤𝑤𝑥) → 𝑣 𝑥)
76ancoms 469 . . . . . . . . . 10 ((𝑤𝑥𝑣𝑤) → 𝑣 𝑥)
85, 7jca 554 . . . . . . . . 9 ((𝑤𝑥𝑣𝑤) → (𝑤𝑥𝑣 𝑥))
98ssopab2i 4968 . . . . . . . 8 {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣 𝑥)}
10 df-xp 5085 . . . . . . . 8 (𝑥 × 𝑥) = {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣 𝑥)}
119, 10sseqtr4i 3622 . . . . . . 7 {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ⊆ (𝑥 × 𝑥)
124, 11ssexi 4768 . . . . . 6 {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ∈ V
13 sseq2 3611 . . . . . . . 8 (𝑦 = {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} → (𝑓𝑦𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}))
14 dmeq 5289 . . . . . . . . 9 (𝑦 = {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} → dom 𝑦 = dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)})
1514fneq2d 5945 . . . . . . . 8 (𝑦 = {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} → (𝑓 Fn dom 𝑦𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}))
1613, 15anbi12d 746 . . . . . . 7 (𝑦 = {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} → ((𝑓𝑦𝑓 Fn dom 𝑦) ↔ (𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)})))
1716exbidv 1847 . . . . . 6 (𝑦 = {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} → (∃𝑓(𝑓𝑦𝑓 Fn dom 𝑦) ↔ ∃𝑓(𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)})))
1812, 17spcv 3288 . . . . 5 (∀𝑦𝑓(𝑓𝑦𝑓 Fn dom 𝑦) → ∃𝑓(𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}))
19 fndm 5953 . . . . . . . . . . . . 13 (𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} → dom 𝑓 = dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)})
20 eleq2 2687 . . . . . . . . . . . . . 14 (dom 𝑓 = dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} → (𝑧 ∈ dom 𝑓𝑧 ∈ dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}))
21 dmopab 5300 . . . . . . . . . . . . . . . 16 dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} = {𝑤 ∣ ∃𝑣(𝑤𝑥𝑣𝑤)}
2221eleq2i 2690 . . . . . . . . . . . . . . 15 (𝑧 ∈ dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ↔ 𝑧 ∈ {𝑤 ∣ ∃𝑣(𝑤𝑥𝑣𝑤)})
23 vex 3192 . . . . . . . . . . . . . . . 16 𝑧 ∈ V
24 elequ1 1994 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑧 → (𝑤𝑥𝑧𝑥))
25 eleq2 2687 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑧 → (𝑣𝑤𝑣𝑧))
2624, 25anbi12d 746 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑧 → ((𝑤𝑥𝑣𝑤) ↔ (𝑧𝑥𝑣𝑧)))
2726exbidv 1847 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑧 → (∃𝑣(𝑤𝑥𝑣𝑤) ↔ ∃𝑣(𝑧𝑥𝑣𝑧)))
2823, 27elab 3337 . . . . . . . . . . . . . . 15 (𝑧 ∈ {𝑤 ∣ ∃𝑣(𝑤𝑥𝑣𝑤)} ↔ ∃𝑣(𝑧𝑥𝑣𝑧))
29 19.42v 1915 . . . . . . . . . . . . . . . 16 (∃𝑣(𝑧𝑥𝑣𝑧) ↔ (𝑧𝑥 ∧ ∃𝑣 𝑣𝑧))
30 n0 3912 . . . . . . . . . . . . . . . . 17 (𝑧 ≠ ∅ ↔ ∃𝑣 𝑣𝑧)
3130anbi2i 729 . . . . . . . . . . . . . . . 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 5951 . . . . . . . . . . . 12 (𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} → Fun 𝑓)
38 funfvima3 6455 . . . . . . . . . . . . 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 2739 . . . . . . . . . . . 12 (𝑧𝑥𝑧 = {𝑢 ∣ (𝑧𝑥𝑢𝑧)})
45 imasng 5451 . . . . . . . . . . . . . 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 3192 . . . . . . . . . . . . . . 15 𝑢 ∈ V
48 elequ1 1994 . . . . . . . . . . . . . . . 16 (𝑣 = 𝑢 → (𝑣𝑧𝑢𝑧))
4948anbi2d 739 . . . . . . . . . . . . . . 15 (𝑣 = 𝑢 → ((𝑧𝑥𝑣𝑧) ↔ (𝑧𝑥𝑢𝑧)))
50 eqid 2621 . . . . . . . . . . . . . . 15 {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} = {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}
5123, 47, 26, 49, 50brab 4963 . . . . . . . . . . . . . 14 (𝑧{⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}𝑢 ↔ (𝑧𝑥𝑢𝑧))
5251abbii 2736 . . . . . . . . . . . . 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 2684 . . . . . . . . . 10 (𝑧𝑥 → ((𝑓𝑧) ∈ ({⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} “ {𝑧}) ↔ (𝑓𝑧) ∈ 𝑧))
5655ad2antrl 763 . . . . . . . . 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 630 . . . . . . 7 ((𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}) → (𝑧𝑥 → (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
5958ralrimiv 2960 . . . . . 6 ((𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤𝑥𝑣𝑤)}) → ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
6059eximi 1759 . . . . 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 1852 . . 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 8894 . . . 4 (∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∃𝑓(𝑓𝑦𝑓 Fn dom 𝑦))
6564alrimiv 1852 . . 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 1478   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wne 2790  wral 2907  Vcvv 3189  wss 3559  c0 3896  {csn 4153   cuni 4407   class class class wbr 4618  {copab 4677  cmpt 4678   × cxp 5077  dom cdm 5079  cima 5082  Fun wfun 5846   Fn wfn 5847  cfv 5852  CHOICEwac 8889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-fv 5860  df-ac 8890
This theorem is referenced by:  dfac4  8896  dfac5  8902  dfac2a  8903  dfac2  8904  dfac8  8908  dfac9  8909  ac4  9248  dfac11  37139
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