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Theorem frec0g 5983
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 4549 . . . . . . . . . 10 dom ∅ = ∅
21biantrur 287 . . . . . . . . 9 (𝑥𝐴 ↔ (dom ∅ = ∅ ∧ 𝑥𝐴))
3 vex 2560 . . . . . . . . . . . . . . . 16 𝑚 ∈ V
4 nsuceq0g 4155 . . . . . . . . . . . . . . . 16 (𝑚 ∈ V → suc 𝑚 ≠ ∅)
53, 4ax-mp 7 . . . . . . . . . . . . . . 15 suc 𝑚 ≠ ∅
65nesymi 2251 . . . . . . . . . . . . . 14 ¬ ∅ = suc 𝑚
71eqeq1i 2047 . . . . . . . . . . . . . 14 (dom ∅ = suc 𝑚 ↔ ∅ = suc 𝑚)
86, 7mtbir 596 . . . . . . . . . . . . 13 ¬ dom ∅ = suc 𝑚
98intnanr 839 . . . . . . . . . . . 12 ¬ (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚)))
109a1i 9 . . . . . . . . . . 11 (𝑚 ∈ ω → ¬ (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚))))
1110nrex 2411 . . . . . . . . . 10 ¬ ∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚)))
1211biorfi 665 . . . . . . . . 9 ((dom ∅ = ∅ ∧ 𝑥𝐴) ↔ ((dom ∅ = ∅ ∧ 𝑥𝐴) ∨ ∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚)))))
13 orcom 647 . . . . . . . . 9 (((dom ∅ = ∅ ∧ 𝑥𝐴) ∨ ∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚)))) ↔ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥𝐴)))
142, 12, 133bitri 195 . . . . . . . 8 (𝑥𝐴 ↔ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥𝐴)))
1514abbii 2153 . . . . . . 7 {𝑥𝑥𝐴} = {𝑥 ∣ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥𝐴))}
16 abid2 2158 . . . . . . 7 {𝑥𝑥𝐴} = 𝐴
1715, 16eqtr3i 2062 . . . . . 6 {𝑥 ∣ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥𝐴))} = 𝐴
18 elex 2566 . . . . . 6 (𝐴𝑉𝐴 ∈ V)
1917, 18syl5eqel 2124 . . . . 5 (𝐴𝑉 → {𝑥 ∣ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥𝐴))} ∈ V)
20 0ex 3884 . . . . . . 7 ∅ ∈ V
21 dmeq 4535 . . . . . . . . . . . . 13 (𝑔 = ∅ → dom 𝑔 = dom ∅)
2221eqeq1d 2048 . . . . . . . . . . . 12 (𝑔 = ∅ → (dom 𝑔 = suc 𝑚 ↔ dom ∅ = suc 𝑚))
23 fveq1 5177 . . . . . . . . . . . . . 14 (𝑔 = ∅ → (𝑔𝑚) = (∅‘𝑚))
2423fveq2d 5182 . . . . . . . . . . . . 13 (𝑔 = ∅ → (𝐹‘(𝑔𝑚)) = (𝐹‘(∅‘𝑚)))
2524eleq2d 2107 . . . . . . . . . . . 12 (𝑔 = ∅ → (𝑥 ∈ (𝐹‘(𝑔𝑚)) ↔ 𝑥 ∈ (𝐹‘(∅‘𝑚))))
2622, 25anbi12d 442 . . . . . . . . . . 11 (𝑔 = ∅ → ((dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ↔ (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚)))))
2726rexbidv 2327 . . . . . . . . . 10 (𝑔 = ∅ → (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ↔ ∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚)))))
2821eqeq1d 2048 . . . . . . . . . . 11 (𝑔 = ∅ → (dom 𝑔 = ∅ ↔ dom ∅ = ∅))
2928anbi1d 438 . . . . . . . . . 10 (𝑔 = ∅ → ((dom 𝑔 = ∅ ∧ 𝑥𝐴) ↔ (dom ∅ = ∅ ∧ 𝑥𝐴)))
3027, 29orbi12d 707 . . . . . . . . 9 (𝑔 = ∅ → ((∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴)) ↔ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥𝐴))))
3130abbidv 2155 . . . . . . . 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 2040 . . . . . . . 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 5248 . . . . . . 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 400 . . . . . 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 2088 . . . . 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 2114 . . 3 (𝐴𝑉 → ((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})‘∅) ∈ V)
38 df-frec 5978 . . . . . 6 frec(𝐹, 𝐴) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})) ↾ ω)
3938fveq1i 5179 . . . . 5 (frec(𝐹, 𝐴)‘∅) = ((recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})) ↾ ω)‘∅)
40 peano1 4317 . . . . . 6 ∅ ∈ ω
41 fvres 5198 . . . . . 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 2060 . . . 4 (frec(𝐹, 𝐴)‘∅) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))}))‘∅)
44 eqid 2040 . . . . 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 5937 . . . 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 2084 . . 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 2072 1 (𝐴𝑉 → (frec(𝐹, 𝐴)‘∅) = 𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 97  wo 629   = wceq 1243  wcel 1393  {cab 2026  wne 2204  wrex 2307  Vcvv 2557  c0 3224  cmpt 3818  suc csuc 4102  ωcom 4313  dom cdm 4345  cres 4347  cfv 4902  recscrecs 5919  freccfrec 5977
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-res 4357  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910  df-recs 5920  df-frec 5978
This theorem is referenced by:  frecrdg  5992  freccl  5993  frec2uz0d  9159  frec2uzrdg  9169  frecuzrdg0  9174
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