Theorem tfrcldm 6572

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 3901	. . 3 ⊢ (𝑌 ∈ ∪ 𝑋 ↔ ∃𝑧(𝑌 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋))
3	1, 2	sylib 122	. 2 ⊢ (𝜑 → ∃𝑧(𝑌 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋))
4		tfrcl.f	. . . 4 ⊢ 𝐹 = recs(𝐺)
5		tfrcl.g	. . . . 5 ⊢ (𝜑 → Fun 𝐺)
6	5	adantr 276	. . . 4 ⊢ ((𝜑 ∧ (𝑌 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋)) → Fun 𝐺)
7		tfrcl.x	. . . . 5 ⊢ (𝜑 → Ord 𝑋)
8	7	adantr 276	. . . 4 ⊢ ((𝜑 ∧ (𝑌 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋)) → Ord 𝑋)
9		tfrcl.ex	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)
10	9	3adant1r 1258	. . . 4 ⊢ (((𝜑 ∧ (𝑌 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋)) ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)
11		feq2 5473	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → (𝑓:𝑎⟶𝑆 ↔ 𝑓:𝑥⟶𝑆))
12		raleq 2731	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → (∀𝑏 ∈ 𝑎 (𝑓‘𝑏) = (𝐺‘(𝑓 ↾ 𝑏)) ↔ ∀𝑏 ∈ 𝑥 (𝑓‘𝑏) = (𝐺‘(𝑓 ↾ 𝑏))))
13	11, 12	anbi12d 473	. . . . . . 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 2769	. . . . . 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 5648	. . . . . . . . . 10 ⊢ (𝑏 = 𝑦 → (𝑓‘𝑏) = (𝑓‘𝑦))
16		reseq2 5014	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑦 → (𝑓 ↾ 𝑏) = (𝑓 ↾ 𝑦))
17	16	fveq2d 5652	. . . . . . . . . 10 ⊢ (𝑏 = 𝑦 → (𝐺‘(𝑓 ↾ 𝑏)) = (𝐺‘(𝑓 ↾ 𝑦)))
18	15, 17	eqeq12d 2246	. . . . . . . . 9 ⊢ (𝑏 = 𝑦 → ((𝑓‘𝑏) = (𝐺‘(𝑓 ↾ 𝑏)) ↔ (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))))
19	18	cbvralv 2768	. . . . . . . 8 ⊢ (∀𝑏 ∈ 𝑥 (𝑓‘𝑏) = (𝐺‘(𝑓 ↾ 𝑏)) ↔ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))
20	19	anbi2i 457	. . . . . . 7 ⊢ ((𝑓:𝑥⟶𝑆 ∧ ∀𝑏 ∈ 𝑥 (𝑓‘𝑏) = (𝐺‘(𝑓 ↾ 𝑏))) ↔ (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))))
21	20	rexbii 2540	. . . . . 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 184	. . . . 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 2347	. . . 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 477	. . . 4 ⊢ (((𝜑 ∧ (𝑌 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋)) ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
26		simprr 533	. . . 4 ⊢ ((𝜑 ∧ (𝑌 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋)) → 𝑧 ∈ 𝑋)
27	4, 6, 8, 10, 23, 25, 26	tfrcllemres 6571	. . 3 ⊢ ((𝜑 ∧ (𝑌 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋)) → 𝑧 ⊆ dom 𝐹)
28		simprl 531	. . 3 ⊢ ((𝜑 ∧ (𝑌 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋)) → 𝑌 ∈ 𝑧)
29	27, 28	sseldd 3229	. 2 ⊢ ((𝜑 ∧ (𝑌 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋)) → 𝑌 ∈ dom 𝐹)
30	3, 29	exlimddv 1947	1 ⊢ (𝜑 → 𝑌 ∈ dom 𝐹)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1005   = wceq 1398  ∃wex 1541   ∈ wcel 2202  {cab 2217  ∀wral 2511  ∃wrex 2512  ∪ cuni 3898  Ord word 4465  suc csuc 4468  dom cdm 4731   ↾ cres 4733  Fun wfun 5327  ⟶wf 5329  ‘cfv 5333  recscrecs 6513

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-recs 6514

This theorem is referenced by:  tfrcl  6573  frecfcllem  6613  frecsuclem  6615