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Theorem tfrcldm 6058
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 3630 . . 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 1163 . . . 4 (((𝜑 ∧ (𝑌𝑧𝑧𝑋)) ∧ 𝑥𝑋𝑓:𝑥𝑆) → (𝐺𝑓) ∈ 𝑆)
11 feq2 5097 . . . . . . . 8 (𝑎 = 𝑥 → (𝑓:𝑎𝑆𝑓:𝑥𝑆))
12 raleq 2555 . . . . . . . 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 2584 . . . . . 6 (∃𝑎𝑋 (𝑓:𝑎𝑆 ∧ ∀𝑏𝑎 (𝑓𝑏) = (𝐺‘(𝑓𝑏))) ↔ ∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑏𝑥 (𝑓𝑏) = (𝐺‘(𝑓𝑏))))
15 fveq2 5251 . . . . . . . . . 10 (𝑏 = 𝑦 → (𝑓𝑏) = (𝑓𝑦))
16 reseq2 4664 . . . . . . . . . . 11 (𝑏 = 𝑦 → (𝑓𝑏) = (𝑓𝑦))
1716fveq2d 5255 . . . . . . . . . 10 (𝑏 = 𝑦 → (𝐺‘(𝑓𝑏)) = (𝐺‘(𝑓𝑦)))
1815, 17eqeq12d 2097 . . . . . . . . 9 (𝑏 = 𝑦 → ((𝑓𝑏) = (𝐺‘(𝑓𝑏)) ↔ (𝑓𝑦) = (𝐺‘(𝑓𝑦))))
1918cbvralv 2583 . . . . . . . 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 2379 . . . . . 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 2198 . . . 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 6057 . . 3 ((𝜑 ∧ (𝑌𝑧𝑧𝑋)) → 𝑧 ⊆ dom 𝐹)
28 simprl 498 . . 3 ((𝜑 ∧ (𝑌𝑧𝑧𝑋)) → 𝑌𝑧)
2927, 28sseldd 3011 . 2 ((𝜑 ∧ (𝑌𝑧𝑧𝑋)) → 𝑌 ∈ dom 𝐹)
303, 29exlimddv 1821 1 (𝜑𝑌 ∈ dom 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 920   = wceq 1285  wex 1422  wcel 1434  {cab 2069  wral 2353  wrex 2354   cuni 3627  Ord word 4152  suc csuc 4155  dom cdm 4399  cres 4401  Fun wfun 4961  wf 4963  cfv 4967  recscrecs 5999
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 3919  ax-sep 3922  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-setind 4315
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 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4083  df-iord 4156  df-on 4158  df-suc 4161  df-xp 4405  df-rel 4406  df-cnv 4407  df-co 4408  df-dm 4409  df-rn 4410  df-res 4411  df-ima 4412  df-iota 4932  df-fun 4969  df-fn 4970  df-f 4971  df-f1 4972  df-fo 4973  df-f1o 4974  df-fv 4975  df-recs 6000
This theorem is referenced by:  tfrcl  6059  frecfcllem  6099  frecsuclem  6101
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