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Theorem frec0g 5922
Description: The initial value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 7-May-2020.)
Assertion
Ref Expression
frec0g (A 𝑉 → (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 4492 . . . . . . . . . 10 dom ∅ = ∅
21biantrur 287 . . . . . . . . 9 (x A ↔ (dom ∅ = ∅ x A))
3 vex 2554 . . . . . . . . . . . . . . . 16 𝑚 V
4 nsuceq0g 4121 . . . . . . . . . . . . . . . 16 (𝑚 V → suc 𝑚 ≠ ∅)
53, 4ax-mp 7 . . . . . . . . . . . . . . 15 suc 𝑚 ≠ ∅
65nesymi 2245 . . . . . . . . . . . . . 14 ¬ ∅ = suc 𝑚
71eqeq1i 2044 . . . . . . . . . . . . . 14 (dom ∅ = suc 𝑚 ↔ ∅ = suc 𝑚)
86, 7mtbir 595 . . . . . . . . . . . . 13 ¬ dom ∅ = suc 𝑚
98intnanr 838 . . . . . . . . . . . 12 ¬ (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚)))
109a1i 9 . . . . . . . . . . 11 (𝑚 𝜔 → ¬ (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚))))
1110nrex 2405 . . . . . . . . . 10 ¬ 𝑚 𝜔 (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚)))
1211biorfi 664 . . . . . . . . 9 ((dom ∅ = ∅ x A) ↔ ((dom ∅ = ∅ x A) 𝑚 𝜔 (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚)))))
13 orcom 646 . . . . . . . . 9 (((dom ∅ = ∅ x A) 𝑚 𝜔 (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚)))) ↔ (𝑚 𝜔 (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚))) (dom ∅ = ∅ x A)))
142, 12, 133bitri 195 . . . . . . . 8 (x A ↔ (𝑚 𝜔 (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚))) (dom ∅ = ∅ x A)))
1514abbii 2150 . . . . . . 7 {xx A} = {x ∣ (𝑚 𝜔 (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚))) (dom ∅ = ∅ x A))}
16 abid2 2155 . . . . . . 7 {xx A} = A
1715, 16eqtr3i 2059 . . . . . 6 {x ∣ (𝑚 𝜔 (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚))) (dom ∅ = ∅ x A))} = A
18 elex 2560 . . . . . 6 (A 𝑉A V)
1917, 18syl5eqel 2121 . . . . 5 (A 𝑉 → {x ∣ (𝑚 𝜔 (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚))) (dom ∅ = ∅ x A))} V)
20 0ex 3875 . . . . . . 7 V
21 dmeq 4478 . . . . . . . . . . . . 13 (g = ∅ → dom g = dom ∅)
2221eqeq1d 2045 . . . . . . . . . . . 12 (g = ∅ → (dom g = suc 𝑚 ↔ dom ∅ = suc 𝑚))
23 fveq1 5120 . . . . . . . . . . . . . 14 (g = ∅ → (g𝑚) = (∅‘𝑚))
2423fveq2d 5125 . . . . . . . . . . . . 13 (g = ∅ → (𝐹‘(g𝑚)) = (𝐹‘(∅‘𝑚)))
2524eleq2d 2104 . . . . . . . . . . . 12 (g = ∅ → (x (𝐹‘(g𝑚)) ↔ x (𝐹‘(∅‘𝑚))))
2622, 25anbi12d 442 . . . . . . . . . . 11 (g = ∅ → ((dom g = suc 𝑚 x (𝐹‘(g𝑚))) ↔ (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚)))))
2726rexbidv 2321 . . . . . . . . . 10 (g = ∅ → (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) ↔ 𝑚 𝜔 (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚)))))
2821eqeq1d 2045 . . . . . . . . . . 11 (g = ∅ → (dom g = ∅ ↔ dom ∅ = ∅))
2928anbi1d 438 . . . . . . . . . 10 (g = ∅ → ((dom g = ∅ x A) ↔ (dom ∅ = ∅ x A)))
3027, 29orbi12d 706 . . . . . . . . 9 (g = ∅ → ((𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A)) ↔ (𝑚 𝜔 (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚))) (dom ∅ = ∅ x A))))
3130abbidv 2152 . . . . . . . 8 (g = ∅ → {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))} = {x ∣ (𝑚 𝜔 (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚))) (dom ∅ = ∅ x A))})
32 eqid 2037 . . . . . . . 8 (g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))}) = (g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})
3331, 32fvmptg 5191 . . . . . . 7 ((∅ V {x ∣ (𝑚 𝜔 (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚))) (dom ∅ = ∅ x A))} V) → ((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})‘∅) = {x ∣ (𝑚 𝜔 (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚))) (dom ∅ = ∅ x A))})
3420, 33mpan 400 . . . . . 6 ({x ∣ (𝑚 𝜔 (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚))) (dom ∅ = ∅ x A))} V → ((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})‘∅) = {x ∣ (𝑚 𝜔 (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚))) (dom ∅ = ∅ x A))})
3534, 17syl6eq 2085 . . . . 5 ({x ∣ (𝑚 𝜔 (dom ∅ = suc 𝑚 x (𝐹‘(∅‘𝑚))) (dom ∅ = ∅ x A))} V → ((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})‘∅) = A)
3619, 35syl 14 . . . 4 (A 𝑉 → ((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})‘∅) = A)
3736, 18eqeltrd 2111 . . 3 (A 𝑉 → ((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})‘∅) V)
38 df-frec 5918 . . . . . 6 frec(𝐹, A) = (recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})) ↾ 𝜔)
3938fveq1i 5122 . . . . 5 (frec(𝐹, A)‘∅) = ((recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})) ↾ 𝜔)‘∅)
40 peano1 4260 . . . . . 6 𝜔
41 fvres 5141 . . . . . 6 (∅ 𝜔 → ((recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})) ↾ 𝜔)‘∅) = (recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))}))‘∅))
4240, 41ax-mp 7 . . . . 5 ((recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})) ↾ 𝜔)‘∅) = (recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))}))‘∅)
4339, 42eqtri 2057 . . . 4 (frec(𝐹, A)‘∅) = (recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))}))‘∅)
44 eqid 2037 . . . . 5 recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})) = recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))}))
4544tfr0 5878 . . . 4 (((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})‘∅) V → (recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))}))‘∅) = ((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})‘∅))
4643, 45syl5eq 2081 . . 3 (((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})‘∅) V → (frec(𝐹, A)‘∅) = ((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})‘∅))
4737, 46syl 14 . 2 (A 𝑉 → (frec(𝐹, A)‘∅) = ((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})‘∅))
4847, 36eqtrd 2069 1 (A 𝑉 → (frec(𝐹, A)‘∅) = A)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   wo 628   = wceq 1242   wcel 1390  {cab 2023  wne 2201  wrex 2301  Vcvv 2551  c0 3218  cmpt 3809  suc csuc 4068  𝜔com 4256  dom cdm 4288  cres 4290  cfv 4845  recscrecs 5860  freccfrec 5917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-res 4300  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853  df-recs 5861  df-frec 5918
This theorem is referenced by:  frecrdg  5931  freccl  5932  frec2uz0d  8846  frec2uzrdg  8856  frecuzrdg0  8861
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