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Theorem frec0g 6397
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 4841 . . . . . . . . . 10 dom ∅ = ∅
21biantrur 303 . . . . . . . . 9 (𝑥𝐴 ↔ (dom ∅ = ∅ ∧ 𝑥𝐴))
3 vex 2740 . . . . . . . . . . . . . . . 16 𝑚 ∈ V
4 nsuceq0g 4418 . . . . . . . . . . . . . . . 16 (𝑚 ∈ V → suc 𝑚 ≠ ∅)
53, 4ax-mp 5 . . . . . . . . . . . . . . 15 suc 𝑚 ≠ ∅
65nesymi 2393 . . . . . . . . . . . . . 14 ¬ ∅ = suc 𝑚
71eqeq1i 2185 . . . . . . . . . . . . . 14 (dom ∅ = suc 𝑚 ↔ ∅ = suc 𝑚)
86, 7mtbir 671 . . . . . . . . . . . . 13 ¬ dom ∅ = suc 𝑚
98intnanr 930 . . . . . . . . . . . 12 ¬ (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚)))
109a1i 9 . . . . . . . . . . 11 (𝑚 ∈ ω → ¬ (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚))))
1110nrex 2569 . . . . . . . . . 10 ¬ ∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚)))
1211biorfi 746 . . . . . . . . 9 ((dom ∅ = ∅ ∧ 𝑥𝐴) ↔ ((dom ∅ = ∅ ∧ 𝑥𝐴) ∨ ∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚)))))
13 orcom 728 . . . . . . . . 9 (((dom ∅ = ∅ ∧ 𝑥𝐴) ∨ ∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚)))) ↔ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥𝐴)))
142, 12, 133bitri 206 . . . . . . . 8 (𝑥𝐴 ↔ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥𝐴)))
1514abbii 2293 . . . . . . 7 {𝑥𝑥𝐴} = {𝑥 ∣ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥𝐴))}
16 abid2 2298 . . . . . . 7 {𝑥𝑥𝐴} = 𝐴
1715, 16eqtr3i 2200 . . . . . 6 {𝑥 ∣ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥𝐴))} = 𝐴
18 elex 2748 . . . . . 6 (𝐴𝑉𝐴 ∈ V)
1917, 18eqeltrid 2264 . . . . 5 (𝐴𝑉 → {𝑥 ∣ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥𝐴))} ∈ V)
20 0ex 4130 . . . . . . 7 ∅ ∈ V
21 dmeq 4827 . . . . . . . . . . . . 13 (𝑔 = ∅ → dom 𝑔 = dom ∅)
2221eqeq1d 2186 . . . . . . . . . . . 12 (𝑔 = ∅ → (dom 𝑔 = suc 𝑚 ↔ dom ∅ = suc 𝑚))
23 fveq1 5514 . . . . . . . . . . . . . 14 (𝑔 = ∅ → (𝑔𝑚) = (∅‘𝑚))
2423fveq2d 5519 . . . . . . . . . . . . 13 (𝑔 = ∅ → (𝐹‘(𝑔𝑚)) = (𝐹‘(∅‘𝑚)))
2524eleq2d 2247 . . . . . . . . . . . 12 (𝑔 = ∅ → (𝑥 ∈ (𝐹‘(𝑔𝑚)) ↔ 𝑥 ∈ (𝐹‘(∅‘𝑚))))
2622, 25anbi12d 473 . . . . . . . . . . 11 (𝑔 = ∅ → ((dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ↔ (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚)))))
2726rexbidv 2478 . . . . . . . . . 10 (𝑔 = ∅ → (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ↔ ∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚)))))
2821eqeq1d 2186 . . . . . . . . . . 11 (𝑔 = ∅ → (dom 𝑔 = ∅ ↔ dom ∅ = ∅))
2928anbi1d 465 . . . . . . . . . 10 (𝑔 = ∅ → ((dom 𝑔 = ∅ ∧ 𝑥𝐴) ↔ (dom ∅ = ∅ ∧ 𝑥𝐴)))
3027, 29orbi12d 793 . . . . . . . . 9 (𝑔 = ∅ → ((∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴)) ↔ (∃𝑚 ∈ ω (dom ∅ = suc 𝑚𝑥 ∈ (𝐹‘(∅‘𝑚))) ∨ (dom ∅ = ∅ ∧ 𝑥𝐴))))
3130abbidv 2295 . . . . . . . 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 2177 . . . . . . . 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 5592 . . . . . . 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 424 . . . . . 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, 17eqtrdi 2226 . . . . 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 2254 . . 3 (𝐴𝑉 → ((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})‘∅) ∈ V)
38 df-frec 6391 . . . . . 6 frec(𝐹, 𝐴) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})) ↾ ω)
3938fveq1i 5516 . . . . 5 (frec(𝐹, 𝐴)‘∅) = ((recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})) ↾ ω)‘∅)
40 peano1 4593 . . . . . 6 ∅ ∈ ω
41 fvres 5539 . . . . . 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 5 . . . . 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 2198 . . . 4 (frec(𝐹, 𝐴)‘∅) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))}))‘∅)
44 eqid 2177 . . . . 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 6323 . . . 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, 45eqtrid 2222 . . 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 2210 1 (𝐴𝑉 → (frec(𝐹, 𝐴)‘∅) = 𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wo 708   = wceq 1353  wcel 2148  {cab 2163  wne 2347  wrex 2456  Vcvv 2737  c0 3422  cmpt 4064  suc csuc 4365  ωcom 4589  dom cdm 4626  cres 4628  cfv 5216  recscrecs 6304  freccfrec 6390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-res 4638  df-iota 5178  df-fun 5218  df-fn 5219  df-fv 5224  df-recs 6305  df-frec 6391
This theorem is referenced by:  frecrdg  6408  frec2uz0d  10398  frec2uzrdg  10408  frecuzrdg0  10412  frecuzrdgg  10415  frecuzrdg0t  10421  seq3val  10457  seqvalcd  10458
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