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Theorem tfr1onlemres 6019
Description: Lemma for tfr1on 6020. Recursion is defined on an ordinal if the characteristic function is defined up to a suitable point. (Contributed by Jim Kingdon, 18-Mar-2022.)
Hypotheses
Ref Expression
tfr1on.f 𝐹 = recs(𝐺)
tfr1on.g (𝜑 → Fun 𝐺)
tfr1on.x (𝜑 → Ord 𝑋)
tfr1on.ex ((𝜑𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)
tfr1onlemsucfn.1 𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
tfr1onlemres.u ((𝜑𝑥 𝑋) → suc 𝑥𝑋)
tfr1onlemres.yx (𝜑𝑌𝑋)
Assertion
Ref Expression
tfr1onlemres (𝜑𝑌 ⊆ dom 𝐹)
Distinct variable groups:   𝑥,𝐴   𝑓,𝐺,𝑥,𝑦   𝑓,𝑋,𝑥   𝑓,𝑌,𝑥   𝜑,𝑓,𝑥
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦,𝑓)   𝐹(𝑥,𝑦,𝑓)   𝑋(𝑦)   𝑌(𝑦)

Proof of Theorem tfr1onlemres
Dummy variables 𝑔 𝑧 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfr1on.x . . . . . . . . . 10 (𝜑 → Ord 𝑋)
21adantr 270 . . . . . . . . 9 ((𝜑𝑧𝑌) → Ord 𝑋)
3 simpr 108 . . . . . . . . . 10 ((𝜑𝑧𝑌) → 𝑧𝑌)
4 tfr1onlemres.yx . . . . . . . . . . 11 (𝜑𝑌𝑋)
54adantr 270 . . . . . . . . . 10 ((𝜑𝑧𝑌) → 𝑌𝑋)
63, 5jca 300 . . . . . . . . 9 ((𝜑𝑧𝑌) → (𝑧𝑌𝑌𝑋))
7 ordtr1 4171 . . . . . . . . 9 (Ord 𝑋 → ((𝑧𝑌𝑌𝑋) → 𝑧𝑋))
82, 6, 7sylc 61 . . . . . . . 8 ((𝜑𝑧𝑌) → 𝑧𝑋)
9 tfr1on.f . . . . . . . . 9 𝐹 = recs(𝐺)
10 tfr1on.g . . . . . . . . 9 (𝜑 → Fun 𝐺)
11 tfr1on.ex . . . . . . . . 9 ((𝜑𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)
12 tfr1onlemsucfn.1 . . . . . . . . 9 𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
13 tfr1onlemres.u . . . . . . . . 9 ((𝜑𝑥 𝑋) → suc 𝑥𝑋)
149, 10, 1, 11, 12, 13tfr1onlemaccex 6018 . . . . . . . 8 ((𝜑𝑧𝑋) → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
158, 14syldan 276 . . . . . . 7 ((𝜑𝑧𝑌) → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
1610ad2antrr 472 . . . . . . . . 9 (((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → Fun 𝐺)
171ad2antrr 472 . . . . . . . . 9 (((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → Ord 𝑋)
18113adant1r 1163 . . . . . . . . . 10 (((𝜑𝑧𝑌) ∧ 𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)
19183adant1r 1163 . . . . . . . . 9 ((((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) ∧ 𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)
204ad2antrr 472 . . . . . . . . 9 (((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → 𝑌𝑋)
213adantr 270 . . . . . . . . 9 (((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → 𝑧𝑌)
2213adantlr 461 . . . . . . . . . 10 (((𝜑𝑧𝑌) ∧ 𝑥 𝑋) → suc 𝑥𝑋)
2322adantlr 461 . . . . . . . . 9 ((((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) ∧ 𝑥 𝑋) → suc 𝑥𝑋)
24 simprl 498 . . . . . . . . 9 (((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → 𝑔 Fn 𝑧)
25 fneq2 5039 . . . . . . . . . . . . 13 (𝑤 = 𝑧 → (𝑔 Fn 𝑤𝑔 Fn 𝑧))
26 raleq 2554 . . . . . . . . . . . . 13 (𝑤 = 𝑧 → (∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢)) ↔ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
2725, 26anbi12d 457 . . . . . . . . . . . 12 (𝑤 = 𝑧 → ((𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))) ↔ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))
2827rspcev 2710 . . . . . . . . . . 11 ((𝑧𝑋 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → ∃𝑤𝑋 (𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
298, 28sylan 277 . . . . . . . . . 10 (((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → ∃𝑤𝑋 (𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
30 vex 2613 . . . . . . . . . . 11 𝑔 ∈ V
3112tfr1onlem3ag 6007 . . . . . . . . . . 11 (𝑔 ∈ V → (𝑔𝐴 ↔ ∃𝑤𝑋 (𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))
3230, 31ax-mp 7 . . . . . . . . . 10 (𝑔𝐴 ↔ ∃𝑤𝑋 (𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
3329, 32sylibr 132 . . . . . . . . 9 (((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → 𝑔𝐴)
349, 16, 17, 19, 12, 20, 21, 23, 24, 33tfr1onlemsucaccv 6011 . . . . . . . 8 (((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝐴)
35 vex 2613 . . . . . . . . . . 11 𝑧 ∈ V
36 fneq2 5039 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (𝑔 Fn 𝑥𝑔 Fn 𝑧))
3736imbi1d 229 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → ((𝑔 Fn 𝑥 → (𝐺𝑔) ∈ V) ↔ (𝑔 Fn 𝑧 → (𝐺𝑔) ∈ V)))
38113expia 1141 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝑋) → (𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
3938alrimiv 1797 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝑋) → ∀𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
40 fneq1 5038 . . . . . . . . . . . . . . . . . . 19 (𝑓 = 𝑔 → (𝑓 Fn 𝑥𝑔 Fn 𝑥))
41 fveq2 5230 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = 𝑔 → (𝐺𝑓) = (𝐺𝑔))
4241eleq1d 2151 . . . . . . . . . . . . . . . . . . 19 (𝑓 = 𝑔 → ((𝐺𝑓) ∈ V ↔ (𝐺𝑔) ∈ V))
4340, 42imbi12d 232 . . . . . . . . . . . . . . . . . 18 (𝑓 = 𝑔 → ((𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V) ↔ (𝑔 Fn 𝑥 → (𝐺𝑔) ∈ V)))
4443spv 1783 . . . . . . . . . . . . . . . . 17 (∀𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V) → (𝑔 Fn 𝑥 → (𝐺𝑔) ∈ V))
4539, 44syl 14 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝑋) → (𝑔 Fn 𝑥 → (𝐺𝑔) ∈ V))
4645ralrimiva 2439 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑥𝑋 (𝑔 Fn 𝑥 → (𝐺𝑔) ∈ V))
4746adantr 270 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑌) → ∀𝑥𝑋 (𝑔 Fn 𝑥 → (𝐺𝑔) ∈ V))
4837, 47, 8rspcdva 2715 . . . . . . . . . . . . 13 ((𝜑𝑧𝑌) → (𝑔 Fn 𝑧 → (𝐺𝑔) ∈ V))
4948imp 122 . . . . . . . . . . . 12 (((𝜑𝑧𝑌) ∧ 𝑔 Fn 𝑧) → (𝐺𝑔) ∈ V)
5024, 49syldan 276 . . . . . . . . . . 11 (((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → (𝐺𝑔) ∈ V)
51 opexg 4011 . . . . . . . . . . 11 ((𝑧 ∈ V ∧ (𝐺𝑔) ∈ V) → ⟨𝑧, (𝐺𝑔)⟩ ∈ V)
5235, 50, 51sylancr 405 . . . . . . . . . 10 (((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → ⟨𝑧, (𝐺𝑔)⟩ ∈ V)
53 snidg 3441 . . . . . . . . . 10 (⟨𝑧, (𝐺𝑔)⟩ ∈ V → ⟨𝑧, (𝐺𝑔)⟩ ∈ {⟨𝑧, (𝐺𝑔)⟩})
54 elun2 3150 . . . . . . . . . 10 (⟨𝑧, (𝐺𝑔)⟩ ∈ {⟨𝑧, (𝐺𝑔)⟩} → ⟨𝑧, (𝐺𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))
5552, 53, 543syl 17 . . . . . . . . 9 (((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → ⟨𝑧, (𝐺𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))
56 opeldmg 4588 . . . . . . . . . 10 ((𝑧 ∈ V ∧ (𝐺𝑔) ∈ V) → (⟨𝑧, (𝐺𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) → 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})))
5735, 50, 56sylancr 405 . . . . . . . . 9 (((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → (⟨𝑧, (𝐺𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) → 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})))
5855, 57mpd 13 . . . . . . . 8 (((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))
59 dmeq 4583 . . . . . . . . . 10 ( = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) → dom = dom (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))
6059eleq2d 2152 . . . . . . . . 9 ( = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) → (𝑧 ∈ dom 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})))
6160rspcev 2710 . . . . . . . 8 (((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝐴𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})) → ∃𝐴 𝑧 ∈ dom )
6234, 58, 61syl2anc 403 . . . . . . 7 (((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → ∃𝐴 𝑧 ∈ dom )
6315, 62exlimddv 1821 . . . . . 6 ((𝜑𝑧𝑌) → ∃𝐴 𝑧 ∈ dom )
64 eliun 3702 . . . . . 6 (𝑧 𝐴 dom ↔ ∃𝐴 𝑧 ∈ dom )
6563, 64sylibr 132 . . . . 5 ((𝜑𝑧𝑌) → 𝑧 𝐴 dom )
6665ex 113 . . . 4 (𝜑 → (𝑧𝑌𝑧 𝐴 dom ))
6766ssrdv 3014 . . 3 (𝜑𝑌 𝐴 dom )
68 dmuni 4593 . . . 4 dom 𝐴 = 𝐴 dom
6912, 1tfr1onlemssrecs 6009 . . . . 5 (𝜑 𝐴 ⊆ recs(𝐺))
70 dmss 4582 . . . . 5 ( 𝐴 ⊆ recs(𝐺) → dom 𝐴 ⊆ dom recs(𝐺))
7169, 70syl 14 . . . 4 (𝜑 → dom 𝐴 ⊆ dom recs(𝐺))
7268, 71syl5eqssr 3053 . . 3 (𝜑 𝐴 dom ⊆ dom recs(𝐺))
7367, 72sstrd 3018 . 2 (𝜑𝑌 ⊆ dom recs(𝐺))
749dmeqi 4584 . 2 dom 𝐹 = dom recs(𝐺)
7573, 74syl6sseqr 3055 1 (𝜑𝑌 ⊆ dom 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 920  wal 1283   = wceq 1285  wex 1422  wcel 1434  {cab 2069  wral 2353  wrex 2354  Vcvv 2610  cun 2980  wss 2982  {csn 3416  cop 3419   cuni 3621   ciun 3698  Ord word 4145  suc csuc 4148  dom cdm 4391  cres 4393  Fun wfun 4946   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-coll 3913  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308
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-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  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-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  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-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-recs 5975
This theorem is referenced by:  tfr1on  6020
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