MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  axdclem2 Structured version   Visualization version   GIF version

Theorem axdclem2 10561
Description: Lemma for axdc 10562. Using the full Axiom of Choice, we can construct a choice function 𝑔 on 𝒫 dom 𝑥. From this, we can build a sequence 𝐹 starting at any value 𝑠 ∈ dom 𝑥 by repeatedly applying 𝑔 to the set (𝐹𝑥) (where 𝑥 is the value from the previous iteration). (Contributed by Mario Carneiro, 25-Jan-2013.)
Hypothesis
Ref Expression
axdclem2.1 𝐹 = (rec((𝑦 ∈ V ↦ (𝑔‘{𝑧𝑦𝑥𝑧})), 𝑠) ↾ ω)
Assertion
Ref Expression
axdclem2 (∃𝑧 𝑠𝑥𝑧 → (ran 𝑥 ⊆ dom 𝑥 → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)))
Distinct variable groups:   𝑓,𝐹,𝑛   𝑦,𝐹,𝑧,𝑛   𝑓,𝑔,𝑥,𝑛   𝑔,𝑠,𝑦,𝑛   𝑧,𝑔   𝑥,𝑦,𝑧
Allowed substitution hints:   𝐹(𝑥,𝑔,𝑠)

Proof of Theorem axdclem2
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 frfnom 8476 . . . . . . 7 (rec((𝑦 ∈ V ↦ (𝑔‘{𝑧𝑦𝑥𝑧})), 𝑠) ↾ ω) Fn ω
2 axdclem2.1 . . . . . . . 8 𝐹 = (rec((𝑦 ∈ V ↦ (𝑔‘{𝑧𝑦𝑥𝑧})), 𝑠) ↾ ω)
32fneq1i 6664 . . . . . . 7 (𝐹 Fn ω ↔ (rec((𝑦 ∈ V ↦ (𝑔‘{𝑧𝑦𝑥𝑧})), 𝑠) ↾ ω) Fn ω)
41, 3mpbir 231 . . . . . 6 𝐹 Fn ω
54a1i 11 . . . . 5 ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → 𝐹 Fn ω)
6 omex 9684 . . . . . 6 ω ∈ V
76a1i 11 . . . . 5 ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ω ∈ V)
85, 7fnexd 7239 . . . 4 ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → 𝐹 ∈ V)
9 fveq2 6905 . . . . . . . 8 (𝑛 = ∅ → (𝐹𝑛) = (𝐹‘∅))
10 suceq 6449 . . . . . . . . 9 (𝑛 = ∅ → suc 𝑛 = suc ∅)
1110fveq2d 6909 . . . . . . . 8 (𝑛 = ∅ → (𝐹‘suc 𝑛) = (𝐹‘suc ∅))
129, 11breq12d 5155 . . . . . . 7 (𝑛 = ∅ → ((𝐹𝑛)𝑥(𝐹‘suc 𝑛) ↔ (𝐹‘∅)𝑥(𝐹‘suc ∅)))
13 fveq2 6905 . . . . . . . 8 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
14 suceq 6449 . . . . . . . . 9 (𝑛 = 𝑘 → suc 𝑛 = suc 𝑘)
1514fveq2d 6909 . . . . . . . 8 (𝑛 = 𝑘 → (𝐹‘suc 𝑛) = (𝐹‘suc 𝑘))
1613, 15breq12d 5155 . . . . . . 7 (𝑛 = 𝑘 → ((𝐹𝑛)𝑥(𝐹‘suc 𝑛) ↔ (𝐹𝑘)𝑥(𝐹‘suc 𝑘)))
17 fveq2 6905 . . . . . . . 8 (𝑛 = suc 𝑘 → (𝐹𝑛) = (𝐹‘suc 𝑘))
18 suceq 6449 . . . . . . . . 9 (𝑛 = suc 𝑘 → suc 𝑛 = suc suc 𝑘)
1918fveq2d 6909 . . . . . . . 8 (𝑛 = suc 𝑘 → (𝐹‘suc 𝑛) = (𝐹‘suc suc 𝑘))
2017, 19breq12d 5155 . . . . . . 7 (𝑛 = suc 𝑘 → ((𝐹𝑛)𝑥(𝐹‘suc 𝑛) ↔ (𝐹‘suc 𝑘)𝑥(𝐹‘suc suc 𝑘)))
212fveq1i 6906 . . . . . . . . . . . . 13 (𝐹‘∅) = ((rec((𝑦 ∈ V ↦ (𝑔‘{𝑧𝑦𝑥𝑧})), 𝑠) ↾ ω)‘∅)
22 fr0g 8477 . . . . . . . . . . . . . 14 (𝑠 ∈ V → ((rec((𝑦 ∈ V ↦ (𝑔‘{𝑧𝑦𝑥𝑧})), 𝑠) ↾ ω)‘∅) = 𝑠)
2322elv 3484 . . . . . . . . . . . . 13 ((rec((𝑦 ∈ V ↦ (𝑔‘{𝑧𝑦𝑥𝑧})), 𝑠) ↾ ω)‘∅) = 𝑠
2421, 23eqtri 2764 . . . . . . . . . . . 12 (𝐹‘∅) = 𝑠
2524breq1i 5149 . . . . . . . . . . 11 ((𝐹‘∅)𝑥𝑧𝑠𝑥𝑧)
2625biimpri 228 . . . . . . . . . 10 (𝑠𝑥𝑧 → (𝐹‘∅)𝑥𝑧)
2726eximi 1834 . . . . . . . . 9 (∃𝑧 𝑠𝑥𝑧 → ∃𝑧(𝐹‘∅)𝑥𝑧)
28 peano1 7911 . . . . . . . . . 10 ∅ ∈ ω
292axdclem 10560 . . . . . . . . . 10 ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥 ∧ ∃𝑧(𝐹‘∅)𝑥𝑧) → (∅ ∈ ω → (𝐹‘∅)𝑥(𝐹‘suc ∅)))
3028, 29mpi 20 . . . . . . . . 9 ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥 ∧ ∃𝑧(𝐹‘∅)𝑥𝑧) → (𝐹‘∅)𝑥(𝐹‘suc ∅))
3127, 30syl3an3 1165 . . . . . . . 8 ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥 ∧ ∃𝑧 𝑠𝑥𝑧) → (𝐹‘∅)𝑥(𝐹‘suc ∅))
32313com23 1126 . . . . . . 7 ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → (𝐹‘∅)𝑥(𝐹‘suc ∅))
33 fvex 6918 . . . . . . . . . . . . . 14 (𝐹𝑘) ∈ V
34 fvex 6918 . . . . . . . . . . . . . 14 (𝐹‘suc 𝑘) ∈ V
3533, 34brelrn 5952 . . . . . . . . . . . . 13 ((𝐹𝑘)𝑥(𝐹‘suc 𝑘) → (𝐹‘suc 𝑘) ∈ ran 𝑥)
36 ssel 3976 . . . . . . . . . . . . 13 (ran 𝑥 ⊆ dom 𝑥 → ((𝐹‘suc 𝑘) ∈ ran 𝑥 → (𝐹‘suc 𝑘) ∈ dom 𝑥))
3735, 36syl5 34 . . . . . . . . . . . 12 (ran 𝑥 ⊆ dom 𝑥 → ((𝐹𝑘)𝑥(𝐹‘suc 𝑘) → (𝐹‘suc 𝑘) ∈ dom 𝑥))
3834eldm 5910 . . . . . . . . . . . 12 ((𝐹‘suc 𝑘) ∈ dom 𝑥 ↔ ∃𝑧(𝐹‘suc 𝑘)𝑥𝑧)
3937, 38imbitrdi 251 . . . . . . . . . . 11 (ran 𝑥 ⊆ dom 𝑥 → ((𝐹𝑘)𝑥(𝐹‘suc 𝑘) → ∃𝑧(𝐹‘suc 𝑘)𝑥𝑧))
4039ad2antll 729 . . . . . . . . . 10 ((𝑘 ∈ ω ∧ (∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥)) → ((𝐹𝑘)𝑥(𝐹‘suc 𝑘) → ∃𝑧(𝐹‘suc 𝑘)𝑥𝑧))
41 peano2 7913 . . . . . . . . . . . . . 14 (𝑘 ∈ ω → suc 𝑘 ∈ ω)
422axdclem 10560 . . . . . . . . . . . . . 14 ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥 ∧ ∃𝑧(𝐹‘suc 𝑘)𝑥𝑧) → (suc 𝑘 ∈ ω → (𝐹‘suc 𝑘)𝑥(𝐹‘suc suc 𝑘)))
4341, 42syl5 34 . . . . . . . . . . . . 13 ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥 ∧ ∃𝑧(𝐹‘suc 𝑘)𝑥𝑧) → (𝑘 ∈ ω → (𝐹‘suc 𝑘)𝑥(𝐹‘suc suc 𝑘)))
44433expia 1121 . . . . . . . . . . . 12 ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥) → (∃𝑧(𝐹‘suc 𝑘)𝑥𝑧 → (𝑘 ∈ ω → (𝐹‘suc 𝑘)𝑥(𝐹‘suc suc 𝑘))))
4544com3r 87 . . . . . . . . . . 11 (𝑘 ∈ ω → ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥) → (∃𝑧(𝐹‘suc 𝑘)𝑥𝑧 → (𝐹‘suc 𝑘)𝑥(𝐹‘suc suc 𝑘))))
4645imp 406 . . . . . . . . . 10 ((𝑘 ∈ ω ∧ (∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥)) → (∃𝑧(𝐹‘suc 𝑘)𝑥𝑧 → (𝐹‘suc 𝑘)𝑥(𝐹‘suc suc 𝑘)))
4740, 46syld 47 . . . . . . . . 9 ((𝑘 ∈ ω ∧ (∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥)) → ((𝐹𝑘)𝑥(𝐹‘suc 𝑘) → (𝐹‘suc 𝑘)𝑥(𝐹‘suc suc 𝑘)))
48473adantr2 1170 . . . . . . . 8 ((𝑘 ∈ ω ∧ (∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥)) → ((𝐹𝑘)𝑥(𝐹‘suc 𝑘) → (𝐹‘suc 𝑘)𝑥(𝐹‘suc suc 𝑘)))
4948ex 412 . . . . . . 7 (𝑘 ∈ ω → ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ((𝐹𝑘)𝑥(𝐹‘suc 𝑘) → (𝐹‘suc 𝑘)𝑥(𝐹‘suc suc 𝑘))))
5012, 16, 20, 32, 49finds2 7921 . . . . . 6 (𝑛 ∈ ω → ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → (𝐹𝑛)𝑥(𝐹‘suc 𝑛)))
5150com12 32 . . . . 5 ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → (𝑛 ∈ ω → (𝐹𝑛)𝑥(𝐹‘suc 𝑛)))
5251ralrimiv 3144 . . . 4 ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∀𝑛 ∈ ω (𝐹𝑛)𝑥(𝐹‘suc 𝑛))
53 fveq1 6904 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝑛) = (𝐹𝑛))
54 fveq1 6904 . . . . . 6 (𝑓 = 𝐹 → (𝑓‘suc 𝑛) = (𝐹‘suc 𝑛))
5553, 54breq12d 5155 . . . . 5 (𝑓 = 𝐹 → ((𝑓𝑛)𝑥(𝑓‘suc 𝑛) ↔ (𝐹𝑛)𝑥(𝐹‘suc 𝑛)))
5655ralbidv 3177 . . . 4 (𝑓 = 𝐹 → (∀𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛) ↔ ∀𝑛 ∈ ω (𝐹𝑛)𝑥(𝐹‘suc 𝑛)))
578, 52, 56spcedv 3597 . . 3 ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛))
58573exp 1119 . 2 (∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) → (∃𝑧 𝑠𝑥𝑧 → (ran 𝑥 ⊆ dom 𝑥 → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛))))
59 vex 3483 . . . . 5 𝑥 ∈ V
6059dmex 7932 . . . 4 dom 𝑥 ∈ V
6160pwex 5379 . . 3 𝒫 dom 𝑥 ∈ V
6261ac4c 10517 . 2 𝑔𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦)
6358, 62exlimiiv 1930 1 (∃𝑧 𝑠𝑥𝑧 → (ran 𝑥 ⊆ dom 𝑥 → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1539  wex 1778  wcel 2107  {cab 2713  wne 2939  wral 3060  Vcvv 3479  wss 3950  c0 4332  𝒫 cpw 4599   class class class wbr 5142  cmpt 5224  dom cdm 5684  ran crn 5685  cres 5686  suc csuc 6385   Fn wfn 6555  cfv 6560  ωcom 7888  reccrdg 8450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682  ax-ac2 10504
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-ac 10157
This theorem is referenced by:  axdc  10562
  Copyright terms: Public domain W3C validator