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Theorem axdclem2 10558
Description: Lemma for axdc 10559. 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 8474 . . . . . . 7 (rec((𝑦 ∈ V ↦ (𝑔‘{𝑧𝑦𝑥𝑧})), 𝑠) ↾ ω) Fn ω
2 axdclem2.1 . . . . . . . 8 𝐹 = (rec((𝑦 ∈ V ↦ (𝑔‘{𝑧𝑦𝑥𝑧})), 𝑠) ↾ ω)
32fneq1i 6666 . . . . . . 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 9681 . . . . . 6 ω ∈ V
76a1i 11 . . . . 5 ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ω ∈ V)
85, 7fnexd 7238 . . . 4 ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → 𝐹 ∈ V)
9 fveq2 6907 . . . . . . . 8 (𝑛 = ∅ → (𝐹𝑛) = (𝐹‘∅))
10 suceq 6452 . . . . . . . . 9 (𝑛 = ∅ → suc 𝑛 = suc ∅)
1110fveq2d 6911 . . . . . . . 8 (𝑛 = ∅ → (𝐹‘suc 𝑛) = (𝐹‘suc ∅))
129, 11breq12d 5161 . . . . . . 7 (𝑛 = ∅ → ((𝐹𝑛)𝑥(𝐹‘suc 𝑛) ↔ (𝐹‘∅)𝑥(𝐹‘suc ∅)))
13 fveq2 6907 . . . . . . . 8 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
14 suceq 6452 . . . . . . . . 9 (𝑛 = 𝑘 → suc 𝑛 = suc 𝑘)
1514fveq2d 6911 . . . . . . . 8 (𝑛 = 𝑘 → (𝐹‘suc 𝑛) = (𝐹‘suc 𝑘))
1613, 15breq12d 5161 . . . . . . 7 (𝑛 = 𝑘 → ((𝐹𝑛)𝑥(𝐹‘suc 𝑛) ↔ (𝐹𝑘)𝑥(𝐹‘suc 𝑘)))
17 fveq2 6907 . . . . . . . 8 (𝑛 = suc 𝑘 → (𝐹𝑛) = (𝐹‘suc 𝑘))
18 suceq 6452 . . . . . . . . 9 (𝑛 = suc 𝑘 → suc 𝑛 = suc suc 𝑘)
1918fveq2d 6911 . . . . . . . 8 (𝑛 = suc 𝑘 → (𝐹‘suc 𝑛) = (𝐹‘suc suc 𝑘))
2017, 19breq12d 5161 . . . . . . 7 (𝑛 = suc 𝑘 → ((𝐹𝑛)𝑥(𝐹‘suc 𝑛) ↔ (𝐹‘suc 𝑘)𝑥(𝐹‘suc suc 𝑘)))
212fveq1i 6908 . . . . . . . . . . . . 13 (𝐹‘∅) = ((rec((𝑦 ∈ V ↦ (𝑔‘{𝑧𝑦𝑥𝑧})), 𝑠) ↾ ω)‘∅)
22 fr0g 8475 . . . . . . . . . . . . . 14 (𝑠 ∈ V → ((rec((𝑦 ∈ V ↦ (𝑔‘{𝑧𝑦𝑥𝑧})), 𝑠) ↾ ω)‘∅) = 𝑠)
2322elv 3483 . . . . . . . . . . . . 13 ((rec((𝑦 ∈ V ↦ (𝑔‘{𝑧𝑦𝑥𝑧})), 𝑠) ↾ ω)‘∅) = 𝑠
2421, 23eqtri 2763 . . . . . . . . . . . 12 (𝐹‘∅) = 𝑠
2524breq1i 5155 . . . . . . . . . . 11 ((𝐹‘∅)𝑥𝑧𝑠𝑥𝑧)
2625biimpri 228 . . . . . . . . . 10 (𝑠𝑥𝑧 → (𝐹‘∅)𝑥𝑧)
2726eximi 1832 . . . . . . . . 9 (∃𝑧 𝑠𝑥𝑧 → ∃𝑧(𝐹‘∅)𝑥𝑧)
28 peano1 7911 . . . . . . . . . 10 ∅ ∈ ω
292axdclem 10557 . . . . . . . . . 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 1164 . . . . . . . 8 ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥 ∧ ∃𝑧 𝑠𝑥𝑧) → (𝐹‘∅)𝑥(𝐹‘suc ∅))
32313com23 1125 . . . . . . 7 ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → (𝐹‘∅)𝑥(𝐹‘suc ∅))
33 fvex 6920 . . . . . . . . . . . . . 14 (𝐹𝑘) ∈ V
34 fvex 6920 . . . . . . . . . . . . . 14 (𝐹‘suc 𝑘) ∈ V
3533, 34brelrn 5956 . . . . . . . . . . . . 13 ((𝐹𝑘)𝑥(𝐹‘suc 𝑘) → (𝐹‘suc 𝑘) ∈ ran 𝑥)
36 ssel 3989 . . . . . . . . . . . . 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 5914 . . . . . . . . . . . 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 10557 . . . . . . . . . . . . . 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 1120 . . . . . . . . . . . 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 1169 . . . . . . . 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 3143 . . . 4 ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∀𝑛 ∈ ω (𝐹𝑛)𝑥(𝐹‘suc 𝑛))
53 fveq1 6906 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝑛) = (𝐹𝑛))
54 fveq1 6906 . . . . . 6 (𝑓 = 𝐹 → (𝑓‘suc 𝑛) = (𝐹‘suc 𝑛))
5553, 54breq12d 5161 . . . . 5 (𝑓 = 𝐹 → ((𝑓𝑛)𝑥(𝑓‘suc 𝑛) ↔ (𝐹𝑛)𝑥(𝐹‘suc 𝑛)))
5655ralbidv 3176 . . . 4 (𝑓 = 𝐹 → (∀𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛) ↔ ∀𝑛 ∈ ω (𝐹𝑛)𝑥(𝐹‘suc 𝑛)))
578, 52, 56spcedv 3598 . . 3 ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ∃𝑧 𝑠𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛))
58573exp 1118 . 2 (∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) → (∃𝑧 𝑠𝑥𝑧 → (ran 𝑥 ⊆ dom 𝑥 → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛))))
59 vex 3482 . . . . 5 𝑥 ∈ V
6059dmex 7932 . . . 4 dom 𝑥 ∈ V
6160pwex 5386 . . 3 𝒫 dom 𝑥 ∈ V
6261ac4c 10514 . 2 𝑔𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦)
6358, 62exlimiiv 1929 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 1537  wex 1776  wcel 2106  {cab 2712  wne 2938  wral 3059  Vcvv 3478  wss 3963  c0 4339  𝒫 cpw 4605   class class class wbr 5148  cmpt 5231  dom cdm 5689  ran crn 5690  cres 5691  suc csuc 6388   Fn wfn 6558  cfv 6563  ωcom 7887  reccrdg 8448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-ac2 10501
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-ac 10154
This theorem is referenced by:  axdc  10559
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