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	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)
tfrcl.u	⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
tfrcl.yx	⊢ (𝜑 → 𝑌 ∈ ∪ 𝑋)

Assertion

Ref	Expression
tfrcldm	⊢ (𝜑 → 𝑌 ∈ 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.

Step	Hyp	Ref	Expression
1		tfrcl.yx	. . 3 ⊢ (𝜑 → 𝑌 ∈ ∪ 𝑋)
2		eluni 3771	. . 3 ⊢ (𝑌 ∈ ∪ 𝑋 ↔ ∃𝑧(𝑌 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋))
3	1, 2	sylib 121	. 2 ⊢ (𝜑 → ∃𝑧(𝑌 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋))
4		tfrcl.f	. . . 4 ⊢ 𝐹 = recs(𝐺)
5		tfrcl.g	. . . . 5 ⊢ (𝜑 → Fun 𝐺)
6	5	adantr 274	. . . 4 ⊢ ((𝜑 ∧ (𝑌 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋)) → Fun 𝐺)
7		tfrcl.x	. . . . 5 ⊢ (𝜑 → Ord 𝑋)
8	7	adantr 274	. . . 4 ⊢ ((𝜑 ∧ (𝑌 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋)) → Ord 𝑋)
9		tfrcl.ex	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)
10	9	3adant1r 1210	. . . 4 ⊢ (((𝜑 ∧ (𝑌 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋)) ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)
11		feq2 5296	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → (𝑓:𝑎⟶𝑆 ↔ 𝑓:𝑥⟶𝑆))
12		raleq 2649	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → (∀𝑏 ∈ 𝑎 (𝑓‘𝑏) = (𝐺‘(𝑓 ↾ 𝑏)) ↔ ∀𝑏 ∈ 𝑥 (𝑓‘𝑏) = (𝐺‘(𝑓 ↾ 𝑏))))
13	11, 12	anbi12d 465	. . . . . . 7 ⊢ (𝑎 = 𝑥 → ((𝑓:𝑎⟶𝑆 ∧ ∀𝑏 ∈ 𝑎 (𝑓‘𝑏) = (𝐺‘(𝑓 ↾ 𝑏))) ↔ (𝑓:𝑥⟶𝑆 ∧ ∀𝑏 ∈ 𝑥 (𝑓‘𝑏) = (𝐺‘(𝑓 ↾ 𝑏)))))
14	13	cbvrexv 2678	. . . . . 6 ⊢ (∃𝑎 ∈ 𝑋 (𝑓:𝑎⟶𝑆 ∧ ∀𝑏 ∈ 𝑎 (𝑓‘𝑏) = (𝐺‘(𝑓 ↾ 𝑏))) ↔ ∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑏 ∈ 𝑥 (𝑓‘𝑏) = (𝐺‘(𝑓 ↾ 𝑏))))
15		fveq2 5461	. . . . . . . . . 10 ⊢ (𝑏 = 𝑦 → (𝑓‘𝑏) = (𝑓‘𝑦))
16		reseq2 4854	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑦 → (𝑓 ↾ 𝑏) = (𝑓 ↾ 𝑦))
17	16	fveq2d 5465	. . . . . . . . . 10 ⊢ (𝑏 = 𝑦 → (𝐺‘(𝑓 ↾ 𝑏)) = (𝐺‘(𝑓 ↾ 𝑦)))
18	15, 17	eqeq12d 2169	. . . . . . . . 9 ⊢ (𝑏 = 𝑦 → ((𝑓‘𝑏) = (𝐺‘(𝑓 ↾ 𝑏)) ↔ (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))))
19	18	cbvralv 2677	. . . . . . . 8 ⊢ (∀𝑏 ∈ 𝑥 (𝑓‘𝑏) = (𝐺‘(𝑓 ↾ 𝑏)) ↔ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))
20	19	anbi2i 453	. . . . . . 7 ⊢ ((𝑓:𝑥⟶𝑆 ∧ ∀𝑏 ∈ 𝑥 (𝑓‘𝑏) = (𝐺‘(𝑓 ↾ 𝑏))) ↔ (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))))
21	20	rexbii 2461	. . . . . 6 ⊢ (∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑏 ∈ 𝑥 (𝑓‘𝑏) = (𝐺‘(𝑓 ↾ 𝑏))) ↔ ∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))))
22	14, 21	bitri 183	. . . . 5 ⊢ (∃𝑎 ∈ 𝑋 (𝑓:𝑎⟶𝑆 ∧ ∀𝑏 ∈ 𝑎 (𝑓‘𝑏) = (𝐺‘(𝑓 ↾ 𝑏))) ↔ ∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))))
23	22	abbii 2270	. . . 4 ⊢ {𝑓 ∣ ∃𝑎 ∈ 𝑋 (𝑓:𝑎⟶𝑆 ∧ ∀𝑏 ∈ 𝑎 (𝑓‘𝑏) = (𝐺‘(𝑓 ↾ 𝑏)))} = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
24		tfrcl.u	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
25	24	adantlr 469	. . . 4 ⊢ (((𝜑 ∧ (𝑌 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋)) ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
26		simprr 522	. . . 4 ⊢ ((𝜑 ∧ (𝑌 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋)) → 𝑧 ∈ 𝑋)
27	4, 6, 8, 10, 23, 25, 26	tfrcllemres 6299	. . 3 ⊢ ((𝜑 ∧ (𝑌 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋)) → 𝑧 ⊆ dom 𝐹)
28		simprl 521	. . 3 ⊢ ((𝜑 ∧ (𝑌 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋)) → 𝑌 ∈ 𝑧)
29	27, 28	sseldd 3125	. 2 ⊢ ((𝜑 ∧ (𝑌 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋)) → 𝑌 ∈ dom 𝐹)
30	3, 29	exlimddv 1875	1 ⊢ (𝜑 → 𝑌 ∈ dom 𝐹)

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