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Theorem tfr0dm 5992
 Description: Transfinite recursion is defined at the empty set. (Contributed by Jim Kingdon, 8-Mar-2022.)
Hypothesis
Ref Expression
tfr.1 𝐹 = recs(𝐺)
Assertion
Ref Expression
tfr0dm ((𝐺‘∅) ∈ 𝑉 → ∅ ∈ dom 𝐹)

Proof of Theorem tfr0dm
Dummy variables 𝑥 𝑓 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ex 3925 . . . . 5 ∅ ∈ V
2 opexg 4011 . . . . 5 ((∅ ∈ V ∧ (𝐺‘∅) ∈ 𝑉) → ⟨∅, (𝐺‘∅)⟩ ∈ V)
31, 2mpan 415 . . . 4 ((𝐺‘∅) ∈ 𝑉 → ⟨∅, (𝐺‘∅)⟩ ∈ V)
4 snidg 3441 . . . 4 (⟨∅, (𝐺‘∅)⟩ ∈ V → ⟨∅, (𝐺‘∅)⟩ ∈ {⟨∅, (𝐺‘∅)⟩})
53, 4syl 14 . . 3 ((𝐺‘∅) ∈ 𝑉 → ⟨∅, (𝐺‘∅)⟩ ∈ {⟨∅, (𝐺‘∅)⟩})
6 fnsng 4997 . . . . 5 ((∅ ∈ V ∧ (𝐺‘∅) ∈ 𝑉) → {⟨∅, (𝐺‘∅)⟩} Fn {∅})
71, 6mpan 415 . . . 4 ((𝐺‘∅) ∈ 𝑉 → {⟨∅, (𝐺‘∅)⟩} Fn {∅})
8 fvsng 5412 . . . . . . 7 ((∅ ∈ V ∧ (𝐺‘∅) ∈ 𝑉) → ({⟨∅, (𝐺‘∅)⟩}‘∅) = (𝐺‘∅))
91, 8mpan 415 . . . . . 6 ((𝐺‘∅) ∈ 𝑉 → ({⟨∅, (𝐺‘∅)⟩}‘∅) = (𝐺‘∅))
10 res0 4664 . . . . . . 7 ({⟨∅, (𝐺‘∅)⟩} ↾ ∅) = ∅
1110fveq2i 5233 . . . . . 6 (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ ∅)) = (𝐺‘∅)
129, 11syl6eqr 2133 . . . . 5 ((𝐺‘∅) ∈ 𝑉 → ({⟨∅, (𝐺‘∅)⟩}‘∅) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ ∅)))
13 fveq2 5230 . . . . . . 7 (𝑦 = ∅ → ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = ({⟨∅, (𝐺‘∅)⟩}‘∅))
14 reseq2 4655 . . . . . . . 8 (𝑦 = ∅ → ({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦) = ({⟨∅, (𝐺‘∅)⟩} ↾ ∅))
1514fveq2d 5234 . . . . . . 7 (𝑦 = ∅ → (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦)) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ ∅)))
1613, 15eqeq12d 2097 . . . . . 6 (𝑦 = ∅ → (({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦)) ↔ ({⟨∅, (𝐺‘∅)⟩}‘∅) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ ∅))))
171, 16ralsn 3454 . . . . 5 (∀𝑦 ∈ {∅} ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦)) ↔ ({⟨∅, (𝐺‘∅)⟩}‘∅) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ ∅)))
1812, 17sylibr 132 . . . 4 ((𝐺‘∅) ∈ 𝑉 → ∀𝑦 ∈ {∅} ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦)))
19 suc0 4194 . . . . . 6 suc ∅ = {∅}
20 0elon 4175 . . . . . . 7 ∅ ∈ On
2120onsuci 4288 . . . . . 6 suc ∅ ∈ On
2219, 21eqeltrri 2156 . . . . 5 {∅} ∈ On
23 fneq2 5039 . . . . . . 7 (𝑥 = {∅} → ({⟨∅, (𝐺‘∅)⟩} Fn 𝑥 ↔ {⟨∅, (𝐺‘∅)⟩} Fn {∅}))
24 raleq 2554 . . . . . . 7 (𝑥 = {∅} → (∀𝑦𝑥 ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦)) ↔ ∀𝑦 ∈ {∅} ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦))))
2523, 24anbi12d 457 . . . . . 6 (𝑥 = {∅} → (({⟨∅, (𝐺‘∅)⟩} Fn 𝑥 ∧ ∀𝑦𝑥 ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦))) ↔ ({⟨∅, (𝐺‘∅)⟩} Fn {∅} ∧ ∀𝑦 ∈ {∅} ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦)))))
2625rspcev 2710 . . . . 5 (({∅} ∈ On ∧ ({⟨∅, (𝐺‘∅)⟩} Fn {∅} ∧ ∀𝑦 ∈ {∅} ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦)))) → ∃𝑥 ∈ On ({⟨∅, (𝐺‘∅)⟩} Fn 𝑥 ∧ ∀𝑦𝑥 ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦))))
2722, 26mpan 415 . . . 4 (({⟨∅, (𝐺‘∅)⟩} Fn {∅} ∧ ∀𝑦 ∈ {∅} ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦))) → ∃𝑥 ∈ On ({⟨∅, (𝐺‘∅)⟩} Fn 𝑥 ∧ ∀𝑦𝑥 ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦))))
287, 18, 27syl2anc 403 . . 3 ((𝐺‘∅) ∈ 𝑉 → ∃𝑥 ∈ On ({⟨∅, (𝐺‘∅)⟩} Fn 𝑥 ∧ ∀𝑦𝑥 ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦))))
29 snexg 3976 . . . . 5 (⟨∅, (𝐺‘∅)⟩ ∈ V → {⟨∅, (𝐺‘∅)⟩} ∈ V)
30 eleq2 2146 . . . . . . 7 (𝑓 = {⟨∅, (𝐺‘∅)⟩} → (⟨∅, (𝐺‘∅)⟩ ∈ 𝑓 ↔ ⟨∅, (𝐺‘∅)⟩ ∈ {⟨∅, (𝐺‘∅)⟩}))
31 fneq1 5038 . . . . . . . . 9 (𝑓 = {⟨∅, (𝐺‘∅)⟩} → (𝑓 Fn 𝑥 ↔ {⟨∅, (𝐺‘∅)⟩} Fn 𝑥))
32 fveq1 5229 . . . . . . . . . . 11 (𝑓 = {⟨∅, (𝐺‘∅)⟩} → (𝑓𝑦) = ({⟨∅, (𝐺‘∅)⟩}‘𝑦))
33 reseq1 4654 . . . . . . . . . . . 12 (𝑓 = {⟨∅, (𝐺‘∅)⟩} → (𝑓𝑦) = ({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦))
3433fveq2d 5234 . . . . . . . . . . 11 (𝑓 = {⟨∅, (𝐺‘∅)⟩} → (𝐺‘(𝑓𝑦)) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦)))
3532, 34eqeq12d 2097 . . . . . . . . . 10 (𝑓 = {⟨∅, (𝐺‘∅)⟩} → ((𝑓𝑦) = (𝐺‘(𝑓𝑦)) ↔ ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦))))
3635ralbidv 2373 . . . . . . . . 9 (𝑓 = {⟨∅, (𝐺‘∅)⟩} → (∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)) ↔ ∀𝑦𝑥 ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦))))
3731, 36anbi12d 457 . . . . . . . 8 (𝑓 = {⟨∅, (𝐺‘∅)⟩} → ((𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))) ↔ ({⟨∅, (𝐺‘∅)⟩} Fn 𝑥 ∧ ∀𝑦𝑥 ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦)))))
3837rexbidv 2374 . . . . . . 7 (𝑓 = {⟨∅, (𝐺‘∅)⟩} → (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))) ↔ ∃𝑥 ∈ On ({⟨∅, (𝐺‘∅)⟩} Fn 𝑥 ∧ ∀𝑦𝑥 ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦)))))
3930, 38anbi12d 457 . . . . . 6 (𝑓 = {⟨∅, (𝐺‘∅)⟩} → ((⟨∅, (𝐺‘∅)⟩ ∈ 𝑓 ∧ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))) ↔ (⟨∅, (𝐺‘∅)⟩ ∈ {⟨∅, (𝐺‘∅)⟩} ∧ ∃𝑥 ∈ On ({⟨∅, (𝐺‘∅)⟩} Fn 𝑥 ∧ ∀𝑦𝑥 ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦))))))
4039spcegv 2695 . . . . 5 ({⟨∅, (𝐺‘∅)⟩} ∈ V → ((⟨∅, (𝐺‘∅)⟩ ∈ {⟨∅, (𝐺‘∅)⟩} ∧ ∃𝑥 ∈ On ({⟨∅, (𝐺‘∅)⟩} Fn 𝑥 ∧ ∀𝑦𝑥 ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦)))) → ∃𝑓(⟨∅, (𝐺‘∅)⟩ ∈ 𝑓 ∧ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))))))
413, 29, 403syl 17 . . . 4 ((𝐺‘∅) ∈ 𝑉 → ((⟨∅, (𝐺‘∅)⟩ ∈ {⟨∅, (𝐺‘∅)⟩} ∧ ∃𝑥 ∈ On ({⟨∅, (𝐺‘∅)⟩} Fn 𝑥 ∧ ∀𝑦𝑥 ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦)))) → ∃𝑓(⟨∅, (𝐺‘∅)⟩ ∈ 𝑓 ∧ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))))))
42 tfr.1 . . . . . 6 𝐹 = recs(𝐺)
4342eleq2i 2149 . . . . 5 (⟨∅, (𝐺‘∅)⟩ ∈ 𝐹 ↔ ⟨∅, (𝐺‘∅)⟩ ∈ recs(𝐺))
44 df-recs 5975 . . . . . 6 recs(𝐺) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
4544eleq2i 2149 . . . . 5 (⟨∅, (𝐺‘∅)⟩ ∈ recs(𝐺) ↔ ⟨∅, (𝐺‘∅)⟩ ∈ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))})
46 eluniab 3633 . . . . 5 (⟨∅, (𝐺‘∅)⟩ ∈ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} ↔ ∃𝑓(⟨∅, (𝐺‘∅)⟩ ∈ 𝑓 ∧ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))))
4743, 45, 463bitri 204 . . . 4 (⟨∅, (𝐺‘∅)⟩ ∈ 𝐹 ↔ ∃𝑓(⟨∅, (𝐺‘∅)⟩ ∈ 𝑓 ∧ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))))
4841, 47syl6ibr 160 . . 3 ((𝐺‘∅) ∈ 𝑉 → ((⟨∅, (𝐺‘∅)⟩ ∈ {⟨∅, (𝐺‘∅)⟩} ∧ ∃𝑥 ∈ On ({⟨∅, (𝐺‘∅)⟩} Fn 𝑥 ∧ ∀𝑦𝑥 ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦)))) → ⟨∅, (𝐺‘∅)⟩ ∈ 𝐹))
495, 28, 48mp2and 424 . 2 ((𝐺‘∅) ∈ 𝑉 → ⟨∅, (𝐺‘∅)⟩ ∈ 𝐹)
50 opeldmg 4588 . . 3 ((∅ ∈ V ∧ (𝐺‘∅) ∈ 𝑉) → (⟨∅, (𝐺‘∅)⟩ ∈ 𝐹 → ∅ ∈ dom 𝐹))
511, 50mpan 415 . 2 ((𝐺‘∅) ∈ 𝑉 → (⟨∅, (𝐺‘∅)⟩ ∈ 𝐹 → ∅ ∈ dom 𝐹))
5249, 51mpd 13 1 ((𝐺‘∅) ∈ 𝑉 → ∅ ∈ dom 𝐹)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 102   = wceq 1285  ∃wex 1422   ∈ wcel 1434  {cab 2069  ∀wral 2353  ∃wrex 2354  Vcvv 2610  ∅c0 3267  {csn 3416  ⟨cop 3419  ∪ cuni 3621  Oncon0 4146  suc csuc 4148  dom cdm 4391   ↾ cres 4393   Fn wfn 4947  ‘cfv 4952  recscrecs 5974 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-tr 3896  df-id 4076  df-iord 4149  df-on 4151  df-suc 4154  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-res 4403  df-iota 4917  df-fun 4954  df-fn 4955  df-fv 4960  df-recs 5975 This theorem is referenced by:  tfr0  5993
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