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Theorem frec0g 5916
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 4491 . . . . . . . . . 10 dom ∅ = ∅
21biantrur 287 . . . . . . . . 9 (x A ↔ (dom ∅ = ∅ x A))
3 vex 2554 . . . . . . . . . . . . . . . 16 𝑚 V
4 nsuceq0g 4120 . . . . . . . . . . . . . . . 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 3874 . . . . . . 7 V
21 dmeq 4477 . . . . . . . . . . . . 13 (g = ∅ → dom g = dom ∅)
2221eqeq1d 2045 . . . . . . . . . . . 12 (g = ∅ → (dom g = suc 𝑚 ↔ dom ∅ = suc 𝑚))
23 fveq1 5118 . . . . . . . . . . . . . 14 (g = ∅ → (g𝑚) = (∅‘𝑚))
2423fveq2d 5123 . . . . . . . . . . . . 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 5189 . . . . . . 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 5915 . . . . . 6 frec(𝐹, A) = (recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})) ↾ 𝜔)
3938fveq1i 5120 . . . . 5 (frec(𝐹, A)‘∅) = ((recs((g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})) ↾ 𝜔)‘∅)
40 peano1 4259 . . . . . 6 𝜔
41 fvres 5139 . . . . . 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 5875 . . . 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 3808  suc csuc 4067  𝜔com 4255  dom cdm 4287  cres 4289  cfv 4844  recscrecs 5857  freccfrec 5914
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 3865  ax-nul 3873  ax-pow 3917  ax-pr 3934  ax-un 4135  ax-setind 4219
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 3352  df-sn 3372  df-pr 3373  df-op 3375  df-uni 3571  df-int 3606  df-iun 3649  df-br 3755  df-opab 3809  df-mpt 3810  df-tr 3845  df-id 4020  df-iord 4068  df-on 4070  df-suc 4073  df-iom 4256  df-xp 4293  df-rel 4294  df-cnv 4295  df-co 4296  df-dm 4297  df-res 4299  df-iota 4809  df-fun 4846  df-fn 4847  df-fv 4852  df-recs 5858  df-frec 5915
This theorem is referenced by:  frecrdg  5925  frec2uz0d  8812
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