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Theorem frec0g 6014
 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 4577 . . . . . . . . . 10 dom ∅ = ∅
21biantrur 291 . . . . . . . . 9 (𝑥𝐴 ↔ (dom ∅ = ∅ ∧ 𝑥𝐴))
3 vex 2577 . . . . . . . . . . . . . . . 16 𝑚 ∈ V
4 nsuceq0g 4183 . . . . . . . . . . . . . . . 16 (𝑚 ∈ V → suc 𝑚 ≠ ∅)
53, 4ax-mp 7 . . . . . . . . . . . . . . 15 suc 𝑚 ≠ ∅
65nesymi 2266 . . . . . . . . . . . . . 14 ¬ ∅ = suc 𝑚
71eqeq1i 2063 . . . . . . . . . . . . . 14 (dom ∅ = suc 𝑚 ↔ ∅ = suc 𝑚)
86, 7mtbir 606 . . . . . . . . . . . . 13 ¬ dom ∅ = suc 𝑚
98intnanr 850 . . . . . . . . . . . 12 ¬ (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚)))
109a1i 9 . . . . . . . . . . 11 (𝑚 ∈ ω → ¬ (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚))))
1110nrex 2428 . . . . . . . . . 10 ¬ ∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚)))
1211biorfi 675 . . . . . . . . 9 ((dom ∅ = ∅ ∧ 𝑥𝐴) ↔ ((dom ∅ = ∅ ∧ 𝑥𝐴) ∨ ∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚)))))
13 orcom 657 . . . . . . . . 9 (((dom ∅ = ∅ ∧ 𝑥𝐴) ∨ ∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚)))) ↔ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥𝐴)))
142, 12, 133bitri 199 . . . . . . . 8 (𝑥𝐴 ↔ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥𝐴)))
1514abbii 2169 . . . . . . 7 {𝑥𝑥𝐴} = {𝑥 ∣ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥𝐴))}
16 abid2 2174 . . . . . . 7 {𝑥𝑥𝐴} = 𝐴
1715, 16eqtr3i 2078 . . . . . 6 {𝑥 ∣ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥𝐴))} = 𝐴
18 elex 2583 . . . . . 6 (𝐴𝑉𝐴 ∈ V)
1917, 18syl5eqel 2140 . . . . 5 (𝐴𝑉 → {𝑥 ∣ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥𝐴))} ∈ V)
20 0ex 3912 . . . . . . 7 ∅ ∈ V
21 dmeq 4563 . . . . . . . . . . . . 13 (𝑔 = ∅ → dom 𝑔 = dom ∅)
2221eqeq1d 2064 . . . . . . . . . . . 12 (𝑔 = ∅ → (dom 𝑔 = suc 𝑚 ↔ dom ∅ = suc 𝑚))
23 fveq1 5205 . . . . . . . . . . . . . 14 (𝑔 = ∅ → (𝑔𝑚) = (∅‘𝑚))
2423fveq2d 5210 . . . . . . . . . . . . 13 (𝑔 = ∅ → (𝐹‘(𝑔𝑚)) = (𝐹‘(∅‘𝑚)))
2524eleq2d 2123 . . . . . . . . . . . 12 (𝑔 = ∅ → (𝑥 ∈ (𝐹‘(𝑔𝑚)) ↔ 𝑥 ∈ (𝐹‘(∅‘𝑚))))
2622, 25anbi12d 450 . . . . . . . . . . 11 (𝑔 = ∅ → ((dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ↔ (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚)))))
2726rexbidv 2344 . . . . . . . . . 10 (𝑔 = ∅ → (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ↔ ∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚)))))
2821eqeq1d 2064 . . . . . . . . . . 11 (𝑔 = ∅ → (dom 𝑔 = ∅ ↔ dom ∅ = ∅))
2928anbi1d 446 . . . . . . . . . 10 (𝑔 = ∅ → ((dom 𝑔 = ∅ ∧ 𝑥𝐴) ↔ (dom ∅ = ∅ ∧ 𝑥𝐴)))
3027, 29orbi12d 717 . . . . . . . . 9 (𝑔 = ∅ → ((∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴)) ↔ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥𝐴))))
3130abbidv 2171 . . . . . . . 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 2056 . . . . . . . 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 5276 . . . . . . 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 408 . . . . . 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, 17syl6eq 2104 . . . . 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 2130 . . 3 (𝐴𝑉 → ((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})‘∅) ∈ V)
38 df-frec 6009 . . . . . 6 frec(𝐹, 𝐴) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})) ↾ ω)
3938fveq1i 5207 . . . . 5 (frec(𝐹, 𝐴)‘∅) = ((recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})) ↾ ω)‘∅)
40 peano1 4345 . . . . . 6 ∅ ∈ ω
41 fvres 5226 . . . . . 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 7 . . . . 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 2076 . . . 4 (frec(𝐹, 𝐴)‘∅) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))}))‘∅)
44 eqid 2056 . . . . 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 5968 . . . 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, 45syl5eq 2100 . . 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 2088 1 (𝐴𝑉 → (frec(𝐹, 𝐴)‘∅) = 𝐴)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 101   ∨ wo 639   = wceq 1259   ∈ wcel 1409  {cab 2042   ≠ wne 2220  ∃wrex 2324  Vcvv 2574  ∅c0 3252   ↦ cmpt 3846  suc csuc 4130  ωcom 4341  dom cdm 4373   ↾ cres 4375  ‘cfv 4930  recscrecs 5950  freccfrec 6008 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-id 4058  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-res 4385  df-iota 4895  df-fun 4932  df-fn 4933  df-fv 4938  df-recs 5951  df-frec 6009 This theorem is referenced by:  frecrdg  6023  freccl  6024  frec2uz0d  9349  frec2uzrdg  9359  frecuzrdg0  9364
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