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Theorem tfrcldm 6364
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 3813 . . 3 (𝑌 𝑋 ↔ ∃𝑧(𝑌𝑧𝑧𝑋))
31, 2sylib 122 . 2 (𝜑 → ∃𝑧(𝑌𝑧𝑧𝑋))
4 tfrcl.f . . . 4 𝐹 = recs(𝐺)
5 tfrcl.g . . . . 5 (𝜑 → Fun 𝐺)
65adantr 276 . . . 4 ((𝜑 ∧ (𝑌𝑧𝑧𝑋)) → Fun 𝐺)
7 tfrcl.x . . . . 5 (𝜑 → Ord 𝑋)
87adantr 276 . . . 4 ((𝜑 ∧ (𝑌𝑧𝑧𝑋)) → Ord 𝑋)
9 tfrcl.ex . . . . 5 ((𝜑𝑥𝑋𝑓:𝑥𝑆) → (𝐺𝑓) ∈ 𝑆)
1093adant1r 1231 . . . 4 (((𝜑 ∧ (𝑌𝑧𝑧𝑋)) ∧ 𝑥𝑋𝑓:𝑥𝑆) → (𝐺𝑓) ∈ 𝑆)
11 feq2 5350 . . . . . . . 8 (𝑎 = 𝑥 → (𝑓:𝑎𝑆𝑓:𝑥𝑆))
12 raleq 2673 . . . . . . . 8 (𝑎 = 𝑥 → (∀𝑏𝑎 (𝑓𝑏) = (𝐺‘(𝑓𝑏)) ↔ ∀𝑏𝑥 (𝑓𝑏) = (𝐺‘(𝑓𝑏))))
1311, 12anbi12d 473 . . . . . . 7 (𝑎 = 𝑥 → ((𝑓:𝑎𝑆 ∧ ∀𝑏𝑎 (𝑓𝑏) = (𝐺‘(𝑓𝑏))) ↔ (𝑓:𝑥𝑆 ∧ ∀𝑏𝑥 (𝑓𝑏) = (𝐺‘(𝑓𝑏)))))
1413cbvrexv 2705 . . . . . 6 (∃𝑎𝑋 (𝑓:𝑎𝑆 ∧ ∀𝑏𝑎 (𝑓𝑏) = (𝐺‘(𝑓𝑏))) ↔ ∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑏𝑥 (𝑓𝑏) = (𝐺‘(𝑓𝑏))))
15 fveq2 5516 . . . . . . . . . 10 (𝑏 = 𝑦 → (𝑓𝑏) = (𝑓𝑦))
16 reseq2 4903 . . . . . . . . . . 11 (𝑏 = 𝑦 → (𝑓𝑏) = (𝑓𝑦))
1716fveq2d 5520 . . . . . . . . . 10 (𝑏 = 𝑦 → (𝐺‘(𝑓𝑏)) = (𝐺‘(𝑓𝑦)))
1815, 17eqeq12d 2192 . . . . . . . . 9 (𝑏 = 𝑦 → ((𝑓𝑏) = (𝐺‘(𝑓𝑏)) ↔ (𝑓𝑦) = (𝐺‘(𝑓𝑦))))
1918cbvralv 2704 . . . . . . . 8 (∀𝑏𝑥 (𝑓𝑏) = (𝐺‘(𝑓𝑏)) ↔ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))
2019anbi2i 457 . . . . . . 7 ((𝑓:𝑥𝑆 ∧ ∀𝑏𝑥 (𝑓𝑏) = (𝐺‘(𝑓𝑏))) ↔ (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))))
2120rexbii 2484 . . . . . 6 (∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑏𝑥 (𝑓𝑏) = (𝐺‘(𝑓𝑏))) ↔ ∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))))
2214, 21bitri 184 . . . . 5 (∃𝑎𝑋 (𝑓:𝑎𝑆 ∧ ∀𝑏𝑎 (𝑓𝑏) = (𝐺‘(𝑓𝑏))) ↔ ∃𝑥𝑋 (𝑓:𝑥𝑆 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))))
2322abbii 2293 . . . 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 477 . . . 4 (((𝜑 ∧ (𝑌𝑧𝑧𝑋)) ∧ 𝑥 𝑋) → suc 𝑥𝑋)
26 simprr 531 . . . 4 ((𝜑 ∧ (𝑌𝑧𝑧𝑋)) → 𝑧𝑋)
274, 6, 8, 10, 23, 25, 26tfrcllemres 6363 . . 3 ((𝜑 ∧ (𝑌𝑧𝑧𝑋)) → 𝑧 ⊆ dom 𝐹)
28 simprl 529 . . 3 ((𝜑 ∧ (𝑌𝑧𝑧𝑋)) → 𝑌𝑧)
2927, 28sseldd 3157 . 2 ((𝜑 ∧ (𝑌𝑧𝑧𝑋)) → 𝑌 ∈ dom 𝐹)
303, 29exlimddv 1898 1 (𝜑𝑌 ∈ dom 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 978   = wceq 1353  wex 1492  wcel 2148  {cab 2163  wral 2455  wrex 2456   cuni 3810  Ord word 4363  suc csuc 4366  dom cdm 4627  cres 4629  Fun wfun 5211  wf 5213  cfv 5217  recscrecs 6305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-recs 6306
This theorem is referenced by:  tfrcl  6365  frecfcllem  6405  frecsuclem  6407
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