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Theorem tfr1onlemres 6254
 Description: Lemma for tfr1on 6255. 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 274 . . . . . . . . 9 ((𝜑𝑧𝑌) → Ord 𝑋)
3 simpr 109 . . . . . . . . . 10 ((𝜑𝑧𝑌) → 𝑧𝑌)
4 tfr1onlemres.yx . . . . . . . . . . 11 (𝜑𝑌𝑋)
54adantr 274 . . . . . . . . . 10 ((𝜑𝑧𝑌) → 𝑌𝑋)
63, 5jca 304 . . . . . . . . 9 ((𝜑𝑧𝑌) → (𝑧𝑌𝑌𝑋))
7 ordtr1 4318 . . . . . . . . 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 6253 . . . . . . . 8 ((𝜑𝑧𝑋) → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
158, 14syldan 280 . . . . . . 7 ((𝜑𝑧𝑌) → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
1610ad2antrr 480 . . . . . . . . 9 (((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → Fun 𝐺)
171ad2antrr 480 . . . . . . . . 9 (((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → Ord 𝑋)
18113adant1r 1210 . . . . . . . . . 10 (((𝜑𝑧𝑌) ∧ 𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)
19183adant1r 1210 . . . . . . . . 9 ((((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) ∧ 𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)
204ad2antrr 480 . . . . . . . . 9 (((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → 𝑌𝑋)
213adantr 274 . . . . . . . . 9 (((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → 𝑧𝑌)
2213adantlr 469 . . . . . . . . . 10 (((𝜑𝑧𝑌) ∧ 𝑥 𝑋) → suc 𝑥𝑋)
2322adantlr 469 . . . . . . . . 9 ((((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) ∧ 𝑥 𝑋) → suc 𝑥𝑋)
24 simprl 521 . . . . . . . . 9 (((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → 𝑔 Fn 𝑧)
25 fneq2 5220 . . . . . . . . . . . . 13 (𝑤 = 𝑧 → (𝑔 Fn 𝑤𝑔 Fn 𝑧))
26 raleq 2629 . . . . . . . . . . . . 13 (𝑤 = 𝑧 → (∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢)) ↔ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
2725, 26anbi12d 465 . . . . . . . . . . . 12 (𝑤 = 𝑧 → ((𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))) ↔ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))
2827rspcev 2793 . . . . . . . . . . 11 ((𝑧𝑋 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → ∃𝑤𝑋 (𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
298, 28sylan 281 . . . . . . . . . 10 (((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → ∃𝑤𝑋 (𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
30 vex 2692 . . . . . . . . . . 11 𝑔 ∈ V
3112tfr1onlem3ag 6242 . . . . . . . . . . 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 133 . . . . . . . . 9 (((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → 𝑔𝐴)
349, 16, 17, 19, 12, 20, 21, 23, 24, 33tfr1onlemsucaccv 6246 . . . . . . . 8 (((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝐴)
35 vex 2692 . . . . . . . . . . 11 𝑧 ∈ V
36 fneq2 5220 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (𝑔 Fn 𝑥𝑔 Fn 𝑧))
3736imbi1d 230 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → ((𝑔 Fn 𝑥 → (𝐺𝑔) ∈ V) ↔ (𝑔 Fn 𝑧 → (𝐺𝑔) ∈ V)))
38113expia 1184 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝑋) → (𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
3938alrimiv 1847 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝑋) → ∀𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
40 fneq1 5219 . . . . . . . . . . . . . . . . . . 19 (𝑓 = 𝑔 → (𝑓 Fn 𝑥𝑔 Fn 𝑥))
41 fveq2 5429 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = 𝑔 → (𝐺𝑓) = (𝐺𝑔))
4241eleq1d 2209 . . . . . . . . . . . . . . . . . . 19 (𝑓 = 𝑔 → ((𝐺𝑓) ∈ V ↔ (𝐺𝑔) ∈ V))
4340, 42imbi12d 233 . . . . . . . . . . . . . . . . . 18 (𝑓 = 𝑔 → ((𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V) ↔ (𝑔 Fn 𝑥 → (𝐺𝑔) ∈ V)))
4443spv 1833 . . . . . . . . . . . . . . . . 17 (∀𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V) → (𝑔 Fn 𝑥 → (𝐺𝑔) ∈ V))
4539, 44syl 14 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝑋) → (𝑔 Fn 𝑥 → (𝐺𝑔) ∈ V))
4645ralrimiva 2508 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑥𝑋 (𝑔 Fn 𝑥 → (𝐺𝑔) ∈ V))
4746adantr 274 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑌) → ∀𝑥𝑋 (𝑔 Fn 𝑥 → (𝐺𝑔) ∈ V))
4837, 47, 8rspcdva 2798 . . . . . . . . . . . . 13 ((𝜑𝑧𝑌) → (𝑔 Fn 𝑧 → (𝐺𝑔) ∈ V))
4948imp 123 . . . . . . . . . . . 12 (((𝜑𝑧𝑌) ∧ 𝑔 Fn 𝑧) → (𝐺𝑔) ∈ V)
5024, 49syldan 280 . . . . . . . . . . 11 (((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → (𝐺𝑔) ∈ V)
51 opexg 4158 . . . . . . . . . . 11 ((𝑧 ∈ V ∧ (𝐺𝑔) ∈ V) → ⟨𝑧, (𝐺𝑔)⟩ ∈ V)
5235, 50, 51sylancr 411 . . . . . . . . . 10 (((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → ⟨𝑧, (𝐺𝑔)⟩ ∈ V)
53 snidg 3561 . . . . . . . . . 10 (⟨𝑧, (𝐺𝑔)⟩ ∈ V → ⟨𝑧, (𝐺𝑔)⟩ ∈ {⟨𝑧, (𝐺𝑔)⟩})
54 elun2 3249 . . . . . . . . . 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 4752 . . . . . . . . . 10 ((𝑧 ∈ V ∧ (𝐺𝑔) ∈ V) → (⟨𝑧, (𝐺𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) → 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})))
5735, 50, 56sylancr 411 . . . . . . . . 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 4747 . . . . . . . . . 10 ( = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) → dom = dom (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}))
6059eleq2d 2210 . . . . . . . . 9 ( = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) → (𝑧 ∈ dom 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})))
6160rspcev 2793 . . . . . . . 8 (((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝐴𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})) → ∃𝐴 𝑧 ∈ dom )
6234, 58, 61syl2anc 409 . . . . . . 7 (((𝜑𝑧𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → ∃𝐴 𝑧 ∈ dom )
6315, 62exlimddv 1871 . . . . . 6 ((𝜑𝑧𝑌) → ∃𝐴 𝑧 ∈ dom )
64 eliun 3825 . . . . . 6 (𝑧 𝐴 dom ↔ ∃𝐴 𝑧 ∈ dom )
6563, 64sylibr 133 . . . . 5 ((𝜑𝑧𝑌) → 𝑧 𝐴 dom )
6665ex 114 . . . 4 (𝜑 → (𝑧𝑌𝑧 𝐴 dom ))
6766ssrdv 3108 . . 3 (𝜑𝑌 𝐴 dom )
68 dmuni 4757 . . . 4 dom 𝐴 = 𝐴 dom
6912, 1tfr1onlemssrecs 6244 . . . . 5 (𝜑 𝐴 ⊆ recs(𝐺))
70 dmss 4746 . . . . 5 ( 𝐴 ⊆ recs(𝐺) → dom 𝐴 ⊆ dom recs(𝐺))
7169, 70syl 14 . . . 4 (𝜑 → dom 𝐴 ⊆ dom recs(𝐺))
7268, 71eqsstrrid 3149 . . 3 (𝜑 𝐴 dom ⊆ dom recs(𝐺))
7367, 72sstrd 3112 . 2 (𝜑𝑌 ⊆ dom recs(𝐺))
749dmeqi 4748 . 2 dom 𝐹 = dom recs(𝐺)
7573, 74sseqtrrdi 3151 1 (𝜑𝑌 ⊆ dom 𝐹)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 963  ∀wal 1330   = wceq 1332  ∃wex 1469   ∈ wcel 1481  {cab 2126  ∀wral 2417  ∃wrex 2418  Vcvv 2689   ∪ cun 3074   ⊆ wss 3076  {csn 3532  ⟨cop 3535  ∪ cuni 3744  ∪ ciun 3821  Ord word 4292  suc csuc 4295  dom cdm 4547   ↾ cres 4549  Fun wfun 5125   Fn wfn 5126  ‘cfv 5131  recscrecs 6209 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-suc 4301  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-recs 6210 This theorem is referenced by:  tfr1on  6255
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