Theorem tfrcldm 6421

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 3842	. . 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 1233	. . . 4 ⊢ (((𝜑 ∧ (𝑌 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋)) ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)
11		feq2 5391	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → (𝑓:𝑎⟶𝑆 ↔ 𝑓:𝑥⟶𝑆))
12		raleq 2693	. . . . . . . 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 2730	. . . . . 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 5558	. . . . . . . . . 10 ⊢ (𝑏 = 𝑦 → (𝑓‘𝑏) = (𝑓‘𝑦))
16		reseq2 4941	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑦 → (𝑓 ↾ 𝑏) = (𝑓 ↾ 𝑦))
17	16	fveq2d 5562	. . . . . . . . . 10 ⊢ (𝑏 = 𝑦 → (𝐺‘(𝑓 ↾ 𝑏)) = (𝐺‘(𝑓 ↾ 𝑦)))
18	15, 17	eqeq12d 2211	. . . . . . . . 9 ⊢ (𝑏 = 𝑦 → ((𝑓‘𝑏) = (𝐺‘(𝑓 ↾ 𝑏)) ↔ (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))))
19	18	cbvralv 2729	. . . . . . . 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 2504	. . . . . 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 2312	. . . 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 531	. . . 4 ⊢ ((𝜑 ∧ (𝑌 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋)) → 𝑧 ∈ 𝑋)
27	4, 6, 8, 10, 23, 25, 26	tfrcllemres 6420	. . 3 ⊢ ((𝜑 ∧ (𝑌 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋)) → 𝑧 ⊆ dom 𝐹)
28		simprl 529	. . 3 ⊢ ((𝜑 ∧ (𝑌 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋)) → 𝑌 ∈ 𝑧)
29	27, 28	sseldd 3184	. 2 ⊢ ((𝜑 ∧ (𝑌 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋)) → 𝑌 ∈ dom 𝐹)
30	3, 29	exlimddv 1913	1 ⊢ (𝜑 → 𝑌 ∈ dom 𝐹)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980   = wceq 1364  ∃wex 1506   ∈ wcel 2167  {cab 2182  ∀wral 2475  ∃wrex 2476  ∪ cuni 3839  Ord word 4397  suc csuc 4400  dom cdm 4663   ↾ cres 4665  Fun wfun 5252  ⟶wf 5254  ‘cfv 5258  recscrecs 6362

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-suc 4406  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-recs 6363

This theorem is referenced by:  tfrcl  6422  frecfcllem  6462  frecsuclem  6464