ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  frec0g GIF version

Theorem frec0g 6417
Description: The initial value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 7-May-2020.)
Assertion
Ref Expression
frec0g (𝐴𝑉 → (frec(𝐹, 𝐴)‘∅) = 𝐴)

Proof of Theorem frec0g
Dummy variables 𝑔 𝑚 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dm0 4856 . . . . . . . . . 10 dom ∅ = ∅
21biantrur 303 . . . . . . . . 9 (𝑥𝐴 ↔ (dom ∅ = ∅ ∧ 𝑥𝐴))
3 vex 2755 . . . . . . . . . . . . . . . 16 𝑚 ∈ V
4 nsuceq0g 4433 . . . . . . . . . . . . . . . 16 (𝑚 ∈ V → suc 𝑚 ≠ ∅)
53, 4ax-mp 5 . . . . . . . . . . . . . . 15 suc 𝑚 ≠ ∅
65nesymi 2406 . . . . . . . . . . . . . 14 ¬ ∅ = suc 𝑚
71eqeq1i 2197 . . . . . . . . . . . . . 14 (dom ∅ = suc 𝑚 ↔ ∅ = suc 𝑚)
86, 7mtbir 672 . . . . . . . . . . . . 13 ¬ dom ∅ = suc 𝑚
98intnanr 931 . . . . . . . . . . . 12 ¬ (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚)))
109a1i 9 . . . . . . . . . . 11 (𝑚 ∈ ω → ¬ (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚))))
1110nrex 2582 . . . . . . . . . 10 ¬ ∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚)))
1211biorfi 747 . . . . . . . . 9 ((dom ∅ = ∅ ∧ 𝑥𝐴) ↔ ((dom ∅ = ∅ ∧ 𝑥𝐴) ∨ ∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚)))))
13 orcom 729 . . . . . . . . 9 (((dom ∅ = ∅ ∧ 𝑥𝐴) ∨ ∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚)))) ↔ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥𝐴)))
142, 12, 133bitri 206 . . . . . . . 8 (𝑥𝐴 ↔ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥𝐴)))
1514abbii 2305 . . . . . . 7 {𝑥𝑥𝐴} = {𝑥 ∣ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥𝐴))}
16 abid2 2310 . . . . . . 7 {𝑥𝑥𝐴} = 𝐴
1715, 16eqtr3i 2212 . . . . . 6 {𝑥 ∣ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥𝐴))} = 𝐴
18 elex 2763 . . . . . 6 (𝐴𝑉𝐴 ∈ V)
1917, 18eqeltrid 2276 . . . . 5 (𝐴𝑉 → {𝑥 ∣ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥𝐴))} ∈ V)
20 0ex 4145 . . . . . . 7 ∅ ∈ V
21 dmeq 4842 . . . . . . . . . . . . 13 (𝑔 = ∅ → dom 𝑔 = dom ∅)
2221eqeq1d 2198 . . . . . . . . . . . 12 (𝑔 = ∅ → (dom 𝑔 = suc 𝑚 ↔ dom ∅ = suc 𝑚))
23 fveq1 5530 . . . . . . . . . . . . . 14 (𝑔 = ∅ → (𝑔𝑚) = (∅‘𝑚))
2423fveq2d 5535 . . . . . . . . . . . . 13 (𝑔 = ∅ → (𝐹‘(𝑔𝑚)) = (𝐹‘(∅‘𝑚)))
2524eleq2d 2259 . . . . . . . . . . . 12 (𝑔 = ∅ → (𝑥 ∈ (𝐹‘(𝑔𝑚)) ↔ 𝑥 ∈ (𝐹‘(∅‘𝑚))))
2622, 25anbi12d 473 . . . . . . . . . . 11 (𝑔 = ∅ → ((dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ↔ (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚)))))
2726rexbidv 2491 . . . . . . . . . 10 (𝑔 = ∅ → (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ↔ ∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚)))))
2821eqeq1d 2198 . . . . . . . . . . 11 (𝑔 = ∅ → (dom 𝑔 = ∅ ↔ dom ∅ = ∅))
2928anbi1d 465 . . . . . . . . . 10 (𝑔 = ∅ → ((dom 𝑔 = ∅ ∧ 𝑥𝐴) ↔ (dom ∅ = ∅ ∧ 𝑥𝐴)))
3027, 29orbi12d 794 . . . . . . . . 9 (𝑔 = ∅ → ((∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴)) ↔ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥𝐴))))
3130abbidv 2307 . . . . . . . 8 (𝑔 = ∅ → {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))} = {𝑥 ∣ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥𝐴))})
32 eqid 2189 . . . . . . . 8 (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))}) = (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})
3331, 32fvmptg 5609 . . . . . . 7 ((∅ ∈ V ∧ {𝑥 ∣ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥𝐴))} ∈ V) → ((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})‘∅) = {𝑥 ∣ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥𝐴))})
3420, 33mpan 424 . . . . . 6 ({𝑥 ∣ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥𝐴))} ∈ V → ((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})‘∅) = {𝑥 ∣ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥𝐴))})
3534, 17eqtrdi 2238 . . . . 5 ({𝑥 ∣ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥𝐴))} ∈ V → ((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})‘∅) = 𝐴)
3619, 35syl 14 . . . 4 (𝐴𝑉 → ((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})‘∅) = 𝐴)
3736, 18eqeltrd 2266 . . 3 (𝐴𝑉 → ((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})‘∅) ∈ V)
38 df-frec 6411 . . . . . 6 frec(𝐹, 𝐴) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})) ↾ ω)
3938fveq1i 5532 . . . . 5 (frec(𝐹, 𝐴)‘∅) = ((recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})) ↾ ω)‘∅)
40 peano1 4608 . . . . . 6 ∅ ∈ ω
41 fvres 5555 . . . . . 6 (∅ ∈ ω → ((recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})) ↾ ω)‘∅) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))}))‘∅))
4240, 41ax-mp 5 . . . . 5 ((recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})) ↾ ω)‘∅) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))}))‘∅)
4339, 42eqtri 2210 . . . 4 (frec(𝐹, 𝐴)‘∅) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))}))‘∅)
44 eqid 2189 . . . . 5 recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})) = recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))}))
4544tfr0 6343 . . . 4 (((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})‘∅) ∈ V → (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))}))‘∅) = ((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})‘∅))
4643, 45eqtrid 2234 . . 3 (((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})‘∅) ∈ V → (frec(𝐹, 𝐴)‘∅) = ((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})‘∅))
4737, 46syl 14 . 2 (𝐴𝑉 → (frec(𝐹, 𝐴)‘∅) = ((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})‘∅))
4847, 36eqtrd 2222 1 (𝐴𝑉 → (frec(𝐹, 𝐴)‘∅) = 𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wo 709   = wceq 1364  wcel 2160  {cab 2175  wne 2360  wrex 2469  Vcvv 2752  c0 3437  cmpt 4079  suc csuc 4380  ωcom 4604  dom cdm 4641  cres 4643  cfv 5232  recscrecs 6324  freccfrec 6410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-iord 4381  df-on 4383  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-res 4653  df-iota 5193  df-fun 5234  df-fn 5235  df-fv 5240  df-recs 6325  df-frec 6411
This theorem is referenced by:  frecrdg  6428  frec2uz0d  10425  frec2uzrdg  10435  frecuzrdg0  10439  frecuzrdgg  10442  frecuzrdg0t  10448  seq3val  10484  seqvalcd  10485
  Copyright terms: Public domain W3C validator