Theorem tfrcldm 6607

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 3922	. . 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 5497	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → (𝑓:𝑎⟶𝑆 ↔ 𝑓:𝑥⟶𝑆))
12		raleq 2743	. . . . . . . 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 2781	. . . . . 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 5675	. . . . . . . . . 10 ⊢ (𝑏 = 𝑦 → (𝑓‘𝑏) = (𝑓‘𝑦))
16		reseq2 5038	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑦 → (𝑓 ↾ 𝑏) = (𝑓 ↾ 𝑦))
17	16	fveq2d 5679	. . . . . . . . . 10 ⊢ (𝑏 = 𝑦 → (𝐺‘(𝑓 ↾ 𝑏)) = (𝐺‘(𝑓 ↾ 𝑦)))
18	15, 17	eqeq12d 2249	. . . . . . . . 9 ⊢ (𝑏 = 𝑦 → ((𝑓‘𝑏) = (𝐺‘(𝑓 ↾ 𝑏)) ↔ (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))))
19	18	cbvralv 2780	. . . . . . . 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 2551	. . . . . 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 2350	. . . 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 6606	. . 3 ⊢ ((𝜑 ∧ (𝑌 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋)) → 𝑧 ⊆ dom 𝐹)
28		simprl 531	. . 3 ⊢ ((𝜑 ∧ (𝑌 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋)) → 𝑌 ∈ 𝑧)
29	27, 28	sseldd 3243	. 2 ⊢ ((𝜑 ∧ (𝑌 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋)) → 𝑌 ∈ dom 𝐹)
30	3, 29	exlimddv 1950	1 ⊢ (𝜑 → 𝑌 ∈ dom 𝐹)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1005   = wceq 1398  ∃wex 1541   ∈ wcel 2205  {cab 2220  ∀wral 2522  ∃wrex 2523  ∪ cuni 3919  Ord word 4488  suc csuc 4491  dom cdm 4754   ↾ cres 4756  Fun wfun 5351  ⟶wf 5353  ‘cfv 5357  recscrecs 6548

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-recs 6549

This theorem is referenced by:  tfrcl  6608  frecfcllem  6648  frecsuclem  6650