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Theorem tfrcldm 6128
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 3656 . . 3 (𝑌 𝑋 ↔ ∃𝑧(𝑌𝑧𝑧𝑋))
31, 2sylib 120 . 2 (𝜑 → ∃𝑧(𝑌𝑧𝑧𝑋))
4 tfrcl.f . . . 4 𝐹 = recs(𝐺)
5 tfrcl.g . . . . 5 (𝜑 → Fun 𝐺)
65adantr 270 . . . 4 ((𝜑 ∧ (𝑌𝑧𝑧𝑋)) → Fun 𝐺)
7 tfrcl.x . . . . 5 (𝜑 → Ord 𝑋)
87adantr 270 . . . 4 ((𝜑 ∧ (𝑌𝑧𝑧𝑋)) → Ord 𝑋)
9 tfrcl.ex . . . . 5 ((𝜑𝑥𝑋𝑓:𝑥𝑆) → (𝐺𝑓) ∈ 𝑆)
1093adant1r 1167 . . . 4 (((𝜑 ∧ (𝑌𝑧𝑧𝑋)) ∧ 𝑥𝑋𝑓:𝑥𝑆) → (𝐺𝑓) ∈ 𝑆)
11 feq2 5146 . . . . . . . 8 (𝑎 = 𝑥 → (𝑓:𝑎𝑆𝑓:𝑥𝑆))
12 raleq 2562 . . . . . . . 8 (𝑎 = 𝑥 → (∀𝑏𝑎 (𝑓𝑏) = (𝐺‘(𝑓𝑏)) ↔ ∀𝑏𝑥 (𝑓𝑏) = (𝐺‘(𝑓𝑏))))
1311, 12anbi12d 457 . . . . . . 7 (𝑎 = 𝑥 → ((𝑓:𝑎𝑆 ∧ ∀𝑏𝑎 (𝑓𝑏) = (𝐺‘(𝑓𝑏))) ↔ (𝑓:𝑥𝑆 ∧ ∀𝑏𝑥 (𝑓𝑏) = (𝐺‘(𝑓𝑏)))))
1413cbvrexv 2591 . . . . . 6 (∃𝑎𝑋 (𝑓:𝑎𝑆 ∧ ∀𝑏𝑎 (𝑓𝑏) = (𝐺‘(𝑓𝑏))) ↔ ∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑏𝑥 (𝑓𝑏) = (𝐺‘(𝑓𝑏))))
15 fveq2 5305 . . . . . . . . . 10 (𝑏 = 𝑦 → (𝑓𝑏) = (𝑓𝑦))
16 reseq2 4708 . . . . . . . . . . 11 (𝑏 = 𝑦 → (𝑓𝑏) = (𝑓𝑦))
1716fveq2d 5309 . . . . . . . . . 10 (𝑏 = 𝑦 → (𝐺‘(𝑓𝑏)) = (𝐺‘(𝑓𝑦)))
1815, 17eqeq12d 2102 . . . . . . . . 9 (𝑏 = 𝑦 → ((𝑓𝑏) = (𝐺‘(𝑓𝑏)) ↔ (𝑓𝑦) = (𝐺‘(𝑓𝑦))))
1918cbvralv 2590 . . . . . . . 8 (∀𝑏𝑥 (𝑓𝑏) = (𝐺‘(𝑓𝑏)) ↔ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))
2019anbi2i 445 . . . . . . 7 ((𝑓:𝑥𝑆 ∧ ∀𝑏𝑥 (𝑓𝑏) = (𝐺‘(𝑓𝑏))) ↔ (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))))
2120rexbii 2385 . . . . . 6 (∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑏𝑥 (𝑓𝑏) = (𝐺‘(𝑓𝑏))) ↔ ∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))))
2214, 21bitri 182 . . . . 5 (∃𝑎𝑋 (𝑓:𝑎𝑆 ∧ ∀𝑏𝑎 (𝑓𝑏) = (𝐺‘(𝑓𝑏))) ↔ ∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))))
2322abbii 2203 . . . 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 461 . . . 4 (((𝜑 ∧ (𝑌𝑧𝑧𝑋)) ∧ 𝑥 𝑋) → suc 𝑥𝑋)
26 simprr 499 . . . 4 ((𝜑 ∧ (𝑌𝑧𝑧𝑋)) → 𝑧𝑋)
274, 6, 8, 10, 23, 25, 26tfrcllemres 6127 . . 3 ((𝜑 ∧ (𝑌𝑧𝑧𝑋)) → 𝑧 ⊆ dom 𝐹)
28 simprl 498 . . 3 ((𝜑 ∧ (𝑌𝑧𝑧𝑋)) → 𝑌𝑧)
2927, 28sseldd 3026 . 2 ((𝜑 ∧ (𝑌𝑧𝑧𝑋)) → 𝑌 ∈ dom 𝐹)
303, 29exlimddv 1826 1 (𝜑𝑌 ∈ dom 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 924   = wceq 1289  wex 1426  wcel 1438  {cab 2074  wral 2359  wrex 2360   cuni 3653  Ord word 4189  suc csuc 4192  dom cdm 4438  cres 4440  Fun wfun 5009  wf 5011  cfv 5015  recscrecs 6069
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-iord 4193  df-on 4195  df-suc 4198  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-recs 6070
This theorem is referenced by:  tfrcl  6129  frecfcllem  6169  frecsuclem  6171
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