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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-bndl 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  8866  frec2uzrdg  8876  frecuzrdg0  8881
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