ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  tfr1onlemres GIF version

Theorem tfr1onlemres 6350
Description: Lemma for tfr1on 6351. 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 276 . . . . . . . . 9 ((𝜑𝑧𝑌) → Ord 𝑋)
3 simpr 110 . . . . . . . . . 10 ((𝜑𝑧𝑌) → 𝑧𝑌)
4 tfr1onlemres.yx . . . . . . . . . . 11 (𝜑𝑌𝑋)
54adantr 276 . . . . . . . . . 10 ((𝜑𝑧𝑌) → 𝑌𝑋)
63, 5jca 306 . . . . . . . . 9 ((𝜑𝑧𝑌) → (𝑧𝑌𝑌𝑋))
7 ordtr1 4389 . . . . . . . . 9 (Ord 𝑋 → ((𝑧𝑌𝑌𝑋) → 𝑧𝑋))
82, 6, 7sylc 62 . . . . . . . 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 6349 . . . . . . . 8 ((𝜑𝑧𝑋) → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
158, 14syldan 282 . . . . . . 7 ((𝜑𝑧𝑌) → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
1610ad2antrr 488 . . . . . . . . 9 (((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → Fun 𝐺)
171ad2antrr 488 . . . . . . . . 9 (((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → Ord 𝑋)
18113adant1r 1231 . . . . . . . . . 10 (((𝜑𝑧𝑌) ∧ 𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)
19183adant1r 1231 . . . . . . . . 9 ((((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) ∧ 𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)
204ad2antrr 488 . . . . . . . . 9 (((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → 𝑌𝑋)
213adantr 276 . . . . . . . . 9 (((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → 𝑧𝑌)
2213adantlr 477 . . . . . . . . . 10 (((𝜑𝑧𝑌) ∧ 𝑥 𝑋) → suc 𝑥𝑋)
2322adantlr 477 . . . . . . . . 9 ((((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) ∧ 𝑥 𝑋) → suc 𝑥𝑋)
24 simprl 529 . . . . . . . . 9 (((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → 𝑔 Fn 𝑧)
25 fneq2 5306 . . . . . . . . . . . . 13 (𝑤 = 𝑧 → (𝑔 Fn 𝑤𝑔 Fn 𝑧))
26 raleq 2673 . . . . . . . . . . . . 13 (𝑤 = 𝑧 → (∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢)) ↔ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
2725, 26anbi12d 473 . . . . . . . . . . . 12 (𝑤 = 𝑧 → ((𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))) ↔ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))
2827rspcev 2842 . . . . . . . . . . 11 ((𝑧𝑋 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → ∃𝑤𝑋 (𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
298, 28sylan 283 . . . . . . . . . 10 (((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → ∃𝑤𝑋 (𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
30 vex 2741 . . . . . . . . . . 11 𝑔 ∈ V
3112tfr1onlem3ag 6338 . . . . . . . . . . 11 (𝑔 ∈ V → (𝑔𝐴 ↔ ∃𝑤𝑋 (𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))
3230, 31ax-mp 5 . . . . . . . . . 10 (𝑔𝐴 ↔ ∃𝑤𝑋 (𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
3329, 32sylibr 134 . . . . . . . . 9 (((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → 𝑔𝐴)
349, 16, 17, 19, 12, 20, 21, 23, 24, 33tfr1onlemsucaccv 6342 . . . . . . . 8 (((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝐴)
35 vex 2741 . . . . . . . . . . 11 𝑧 ∈ V
36 fneq2 5306 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (𝑔 Fn 𝑥𝑔 Fn 𝑧))
3736imbi1d 231 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → ((𝑔 Fn 𝑥 → (𝐺𝑔) ∈ V) ↔ (𝑔 Fn 𝑧 → (𝐺𝑔) ∈ V)))
38113expia 1205 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝑋) → (𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
3938alrimiv 1874 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝑋) → ∀𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
40 fneq1 5305 . . . . . . . . . . . . . . . . . . 19 (𝑓 = 𝑔 → (𝑓 Fn 𝑥𝑔 Fn 𝑥))
41 fveq2 5516 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = 𝑔 → (𝐺𝑓) = (𝐺𝑔))
4241eleq1d 2246 . . . . . . . . . . . . . . . . . . 19 (𝑓 = 𝑔 → ((𝐺𝑓) ∈ V ↔ (𝐺𝑔) ∈ V))
4340, 42imbi12d 234 . . . . . . . . . . . . . . . . . 18 (𝑓 = 𝑔 → ((𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V) ↔ (𝑔 Fn 𝑥 → (𝐺𝑔) ∈ V)))
4443spv 1860 . . . . . . . . . . . . . . . . 17 (∀𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V) → (𝑔 Fn 𝑥 → (𝐺𝑔) ∈ V))
4539, 44syl 14 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝑋) → (𝑔 Fn 𝑥 → (𝐺𝑔) ∈ V))
4645ralrimiva 2550 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑥𝑋 (𝑔 Fn 𝑥 → (𝐺𝑔) ∈ V))
4746adantr 276 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑌) → ∀𝑥𝑋 (𝑔 Fn 𝑥 → (𝐺𝑔) ∈ V))
4837, 47, 8rspcdva 2847 . . . . . . . . . . . . 13 ((𝜑𝑧𝑌) → (𝑔 Fn 𝑧 → (𝐺𝑔) ∈ V))
4948imp 124 . . . . . . . . . . . 12 (((𝜑𝑧𝑌) ∧ 𝑔 Fn 𝑧) → (𝐺𝑔) ∈ V)
5024, 49syldan 282 . . . . . . . . . . 11 (((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → (𝐺𝑔) ∈ V)
51 opexg 4229 . . . . . . . . . . 11 ((𝑧 ∈ V ∧ (𝐺𝑔) ∈ V) → ⟨𝑧, (𝐺𝑔)⟩ ∈ V)
5235, 50, 51sylancr 414 . . . . . . . . . 10 (((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → ⟨𝑧, (𝐺𝑔)⟩ ∈ V)
53 snidg 3622 . . . . . . . . . 10 (⟨𝑧, (𝐺𝑔)⟩ ∈ V → ⟨𝑧, (𝐺𝑔)⟩ ∈ {⟨𝑧, (𝐺𝑔)⟩})
54 elun2 3304 . . . . . . . . . 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 4833 . . . . . . . . . 10 ((𝑧 ∈ V ∧ (𝐺𝑔) ∈ V) → (⟨𝑧, (𝐺𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) → 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})))
5735, 50, 56sylancr 414 . . . . . . . . 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 4828 . . . . . . . . . 10 ( = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) → dom = dom (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))
6059eleq2d 2247 . . . . . . . . 9 ( = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) → (𝑧 ∈ dom 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})))
6160rspcev 2842 . . . . . . . 8 (((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝐴𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})) → ∃𝐴 𝑧 ∈ dom )
6234, 58, 61syl2anc 411 . . . . . . 7 (((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → ∃𝐴 𝑧 ∈ dom )
6315, 62exlimddv 1898 . . . . . 6 ((𝜑𝑧𝑌) → ∃𝐴 𝑧 ∈ dom )
64 eliun 3891 . . . . . 6 (𝑧 𝐴 dom ↔ ∃𝐴 𝑧 ∈ dom )
6563, 64sylibr 134 . . . . 5 ((𝜑𝑧𝑌) → 𝑧 𝐴 dom )
6665ex 115 . . . 4 (𝜑 → (𝑧𝑌𝑧 𝐴 dom ))
6766ssrdv 3162 . . 3 (𝜑𝑌 𝐴 dom )
68 dmuni 4838 . . . 4 dom 𝐴 = 𝐴 dom
6912, 1tfr1onlemssrecs 6340 . . . . 5 (𝜑 𝐴 ⊆ recs(𝐺))
70 dmss 4827 . . . . 5 ( 𝐴 ⊆ recs(𝐺) → dom 𝐴 ⊆ dom recs(𝐺))
7169, 70syl 14 . . . 4 (𝜑 → dom 𝐴 ⊆ dom recs(𝐺))
7268, 71eqsstrrid 3203 . . 3 (𝜑 𝐴 dom ⊆ dom recs(𝐺))
7367, 72sstrd 3166 . 2 (𝜑𝑌 ⊆ dom recs(𝐺))
749dmeqi 4829 . 2 dom 𝐹 = dom recs(𝐺)
7573, 74sseqtrrdi 3205 1 (𝜑𝑌 ⊆ dom 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 978  wal 1351   = wceq 1353  wex 1492  wcel 2148  {cab 2163  wral 2455  wrex 2456  Vcvv 2738  cun 3128  wss 3130  {csn 3593  cop 3596   cuni 3810   ciun 3887  Ord word 4363  suc csuc 4366  dom cdm 4627  cres 4629  Fun wfun 5211   Fn wfn 5212  cfv 5217  recscrecs 6305
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-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537
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-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-recs 6306
This theorem is referenced by:  tfr1on  6351
  Copyright terms: Public domain W3C validator