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Theorem tfrcldm 6300
 Description: Recursion is defined on an ordinal if the characteristic function satisfies a closure hypothesis up to a suitable point. (Contributed by Jim Kingdon, 26-Mar-2022.)
Hypotheses
Ref Expression
tfrcl.f 𝐹 = recs(𝐺)
tfrcl.g (𝜑 → Fun 𝐺)
tfrcl.x (𝜑 → Ord 𝑋)
tfrcl.ex ((𝜑𝑥𝑋𝑓:𝑥𝑆) → (𝐺𝑓) ∈ 𝑆)
tfrcl.u ((𝜑𝑥 𝑋) → suc 𝑥𝑋)
tfrcl.yx (𝜑𝑌 𝑋)
Assertion
Ref Expression
tfrcldm (𝜑𝑌 ∈ dom 𝐹)
Distinct variable groups:   𝑓,𝐺,𝑥   𝑆,𝑓,𝑥   𝑓,𝑋,𝑥   𝑓,𝑌,𝑥   𝜑,𝑓,𝑥
Allowed substitution hints:   𝐹(𝑥,𝑓)

Proof of Theorem tfrcldm
Dummy variables 𝑧 𝑎 𝑏 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrcl.yx . . 3 (𝜑𝑌 𝑋)
2 eluni 3771 . . 3 (𝑌 𝑋 ↔ ∃𝑧(𝑌𝑧𝑧𝑋))
31, 2sylib 121 . 2 (𝜑 → ∃𝑧(𝑌𝑧𝑧𝑋))
4 tfrcl.f . . . 4 𝐹 = recs(𝐺)
5 tfrcl.g . . . . 5 (𝜑 → Fun 𝐺)
65adantr 274 . . . 4 ((𝜑 ∧ (𝑌𝑧𝑧𝑋)) → Fun 𝐺)
7 tfrcl.x . . . . 5 (𝜑 → Ord 𝑋)
87adantr 274 . . . 4 ((𝜑 ∧ (𝑌𝑧𝑧𝑋)) → Ord 𝑋)
9 tfrcl.ex . . . . 5 ((𝜑𝑥𝑋𝑓:𝑥𝑆) → (𝐺𝑓) ∈ 𝑆)
1093adant1r 1210 . . . 4 (((𝜑 ∧ (𝑌𝑧𝑧𝑋)) ∧ 𝑥𝑋𝑓:𝑥𝑆) → (𝐺𝑓) ∈ 𝑆)
11 feq2 5296 . . . . . . . 8 (𝑎 = 𝑥 → (𝑓:𝑎𝑆𝑓:𝑥𝑆))
12 raleq 2649 . . . . . . . 8 (𝑎 = 𝑥 → (∀𝑏𝑎 (𝑓𝑏) = (𝐺‘(𝑓𝑏)) ↔ ∀𝑏𝑥 (𝑓𝑏) = (𝐺‘(𝑓𝑏))))
1311, 12anbi12d 465 . . . . . . 7 (𝑎 = 𝑥 → ((𝑓:𝑎𝑆 ∧ ∀𝑏𝑎 (𝑓𝑏) = (𝐺‘(𝑓𝑏))) ↔ (𝑓:𝑥𝑆 ∧ ∀𝑏𝑥 (𝑓𝑏) = (𝐺‘(𝑓𝑏)))))
1413cbvrexv 2678 . . . . . 6 (∃𝑎𝑋 (𝑓:𝑎𝑆 ∧ ∀𝑏𝑎 (𝑓𝑏) = (𝐺‘(𝑓𝑏))) ↔ ∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑏𝑥 (𝑓𝑏) = (𝐺‘(𝑓𝑏))))
15 fveq2 5461 . . . . . . . . . 10 (𝑏 = 𝑦 → (𝑓𝑏) = (𝑓𝑦))
16 reseq2 4854 . . . . . . . . . . 11 (𝑏 = 𝑦 → (𝑓𝑏) = (𝑓𝑦))
1716fveq2d 5465 . . . . . . . . . 10 (𝑏 = 𝑦 → (𝐺‘(𝑓𝑏)) = (𝐺‘(𝑓𝑦)))
1815, 17eqeq12d 2169 . . . . . . . . 9 (𝑏 = 𝑦 → ((𝑓𝑏) = (𝐺‘(𝑓𝑏)) ↔ (𝑓𝑦) = (𝐺‘(𝑓𝑦))))
1918cbvralv 2677 . . . . . . . 8 (∀𝑏𝑥 (𝑓𝑏) = (𝐺‘(𝑓𝑏)) ↔ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))
2019anbi2i 453 . . . . . . 7 ((𝑓:𝑥𝑆 ∧ ∀𝑏𝑥 (𝑓𝑏) = (𝐺‘(𝑓𝑏))) ↔ (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))))
2120rexbii 2461 . . . . . 6 (∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑏𝑥 (𝑓𝑏) = (𝐺‘(𝑓𝑏))) ↔ ∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))))
2214, 21bitri 183 . . . . 5 (∃𝑎𝑋 (𝑓:𝑎𝑆 ∧ ∀𝑏𝑎 (𝑓𝑏) = (𝐺‘(𝑓𝑏))) ↔ ∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))))
2322abbii 2270 . . . 4 {𝑓 ∣ ∃𝑎𝑋 (𝑓:𝑎𝑆 ∧ ∀𝑏𝑎 (𝑓𝑏) = (𝐺‘(𝑓𝑏)))} = {𝑓 ∣ ∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
24 tfrcl.u . . . . 5 ((𝜑𝑥 𝑋) → suc 𝑥𝑋)
2524adantlr 469 . . . 4 (((𝜑 ∧ (𝑌𝑧𝑧𝑋)) ∧ 𝑥 𝑋) → suc 𝑥𝑋)
26 simprr 522 . . . 4 ((𝜑 ∧ (𝑌𝑧𝑧𝑋)) → 𝑧𝑋)
274, 6, 8, 10, 23, 25, 26tfrcllemres 6299 . . 3 ((𝜑 ∧ (𝑌𝑧𝑧𝑋)) → 𝑧 ⊆ dom 𝐹)
28 simprl 521 . . 3 ((𝜑 ∧ (𝑌𝑧𝑧𝑋)) → 𝑌𝑧)
2927, 28sseldd 3125 . 2 ((𝜑 ∧ (𝑌𝑧𝑧𝑋)) → 𝑌 ∈ dom 𝐹)
303, 29exlimddv 1875 1 (𝜑𝑌 ∈ dom 𝐹)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 963   = wceq 1332  ∃wex 1469   ∈ wcel 2125  {cab 2140  ∀wral 2432  ∃wrex 2433  ∪ cuni 3768  Ord word 4317  suc csuc 4320  dom cdm 4579   ↾ cres 4581  Fun wfun 5157  ⟶wf 5159  ‘cfv 5163  recscrecs 6241 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 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-coll 4075  ax-sep 4078  ax-pow 4130  ax-pr 4164  ax-un 4388  ax-setind 4490 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-ral 2437  df-rex 2438  df-reu 2439  df-rab 2441  df-v 2711  df-sbc 2934  df-csb 3028  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-nul 3391  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-iun 3847  df-br 3962  df-opab 4022  df-mpt 4023  df-tr 4059  df-id 4248  df-iord 4321  df-on 4323  df-suc 4326  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-res 4591  df-ima 4592  df-iota 5128  df-fun 5165  df-fn 5166  df-f 5167  df-f1 5168  df-fo 5169  df-f1o 5170  df-fv 5171  df-recs 6242 This theorem is referenced by:  tfrcl  6301  frecfcllem  6341  frecsuclem  6343
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