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Theorem tfrcldm 6226
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 3707 . . 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 272 . . . 4 ((𝜑 ∧ (𝑌𝑧𝑧𝑋)) → Fun 𝐺)
7 tfrcl.x . . . . 5 (𝜑 → Ord 𝑋)
87adantr 272 . . . 4 ((𝜑 ∧ (𝑌𝑧𝑧𝑋)) → Ord 𝑋)
9 tfrcl.ex . . . . 5 ((𝜑𝑥𝑋𝑓:𝑥𝑆) → (𝐺𝑓) ∈ 𝑆)
1093adant1r 1192 . . . 4 (((𝜑 ∧ (𝑌𝑧𝑧𝑋)) ∧ 𝑥𝑋𝑓:𝑥𝑆) → (𝐺𝑓) ∈ 𝑆)
11 feq2 5224 . . . . . . . 8 (𝑎 = 𝑥 → (𝑓:𝑎𝑆𝑓:𝑥𝑆))
12 raleq 2601 . . . . . . . 8 (𝑎 = 𝑥 → (∀𝑏𝑎 (𝑓𝑏) = (𝐺‘(𝑓𝑏)) ↔ ∀𝑏𝑥 (𝑓𝑏) = (𝐺‘(𝑓𝑏))))
1311, 12anbi12d 462 . . . . . . 7 (𝑎 = 𝑥 → ((𝑓:𝑎𝑆 ∧ ∀𝑏𝑎 (𝑓𝑏) = (𝐺‘(𝑓𝑏))) ↔ (𝑓:𝑥𝑆 ∧ ∀𝑏𝑥 (𝑓𝑏) = (𝐺‘(𝑓𝑏)))))
1413cbvrexv 2630 . . . . . 6 (∃𝑎𝑋 (𝑓:𝑎𝑆 ∧ ∀𝑏𝑎 (𝑓𝑏) = (𝐺‘(𝑓𝑏))) ↔ ∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑏𝑥 (𝑓𝑏) = (𝐺‘(𝑓𝑏))))
15 fveq2 5387 . . . . . . . . . 10 (𝑏 = 𝑦 → (𝑓𝑏) = (𝑓𝑦))
16 reseq2 4782 . . . . . . . . . . 11 (𝑏 = 𝑦 → (𝑓𝑏) = (𝑓𝑦))
1716fveq2d 5391 . . . . . . . . . 10 (𝑏 = 𝑦 → (𝐺‘(𝑓𝑏)) = (𝐺‘(𝑓𝑦)))
1815, 17eqeq12d 2130 . . . . . . . . 9 (𝑏 = 𝑦 → ((𝑓𝑏) = (𝐺‘(𝑓𝑏)) ↔ (𝑓𝑦) = (𝐺‘(𝑓𝑦))))
1918cbvralv 2629 . . . . . . . 8 (∀𝑏𝑥 (𝑓𝑏) = (𝐺‘(𝑓𝑏)) ↔ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))
2019anbi2i 450 . . . . . . 7 ((𝑓:𝑥𝑆 ∧ ∀𝑏𝑥 (𝑓𝑏) = (𝐺‘(𝑓𝑏))) ↔ (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))))
2120rexbii 2417 . . . . . 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 2231 . . . 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 466 . . . 4 (((𝜑 ∧ (𝑌𝑧𝑧𝑋)) ∧ 𝑥 𝑋) → suc 𝑥𝑋)
26 simprr 504 . . . 4 ((𝜑 ∧ (𝑌𝑧𝑧𝑋)) → 𝑧𝑋)
274, 6, 8, 10, 23, 25, 26tfrcllemres 6225 . . 3 ((𝜑 ∧ (𝑌𝑧𝑧𝑋)) → 𝑧 ⊆ dom 𝐹)
28 simprl 503 . . 3 ((𝜑 ∧ (𝑌𝑧𝑧𝑋)) → 𝑌𝑧)
2927, 28sseldd 3066 . 2 ((𝜑 ∧ (𝑌𝑧𝑧𝑋)) → 𝑌 ∈ dom 𝐹)
303, 29exlimddv 1852 1 (𝜑𝑌 ∈ dom 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 945   = wceq 1314  wex 1451  wcel 1463  {cab 2101  wral 2391  wrex 2392   cuni 3704  Ord word 4252  suc csuc 4255  dom cdm 4507  cres 4509  Fun wfun 5085  wf 5087  cfv 5091  recscrecs 6167
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-recs 6168
This theorem is referenced by:  tfrcl  6227  frecfcllem  6267  frecsuclem  6269
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