Theorem tfrcldm 6515

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 3891	. . 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 1255	. . . 4 ⊢ (((𝜑 ∧ (𝑌 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋)) ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)
11		feq2 5457	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → (𝑓:𝑎⟶𝑆 ↔ 𝑓:𝑥⟶𝑆))
12		raleq 2728	. . . . . . . 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 2766	. . . . . 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 5629	. . . . . . . . . 10 ⊢ (𝑏 = 𝑦 → (𝑓‘𝑏) = (𝑓‘𝑦))
16		reseq2 5000	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑦 → (𝑓 ↾ 𝑏) = (𝑓 ↾ 𝑦))
17	16	fveq2d 5633	. . . . . . . . . 10 ⊢ (𝑏 = 𝑦 → (𝐺‘(𝑓 ↾ 𝑏)) = (𝐺‘(𝑓 ↾ 𝑦)))
18	15, 17	eqeq12d 2244	. . . . . . . . 9 ⊢ (𝑏 = 𝑦 → ((𝑓‘𝑏) = (𝐺‘(𝑓 ↾ 𝑏)) ↔ (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))))
19	18	cbvralv 2765	. . . . . . . 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 2537	. . . . . 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 2345	. . . 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 6514	. . 3 ⊢ ((𝜑 ∧ (𝑌 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋)) → 𝑧 ⊆ dom 𝐹)
28		simprl 529	. . 3 ⊢ ((𝜑 ∧ (𝑌 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋)) → 𝑌 ∈ 𝑧)
29	27, 28	sseldd 3225	. 2 ⊢ ((𝜑 ∧ (𝑌 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋)) → 𝑌 ∈ dom 𝐹)
30	3, 29	exlimddv 1945	1 ⊢ (𝜑 → 𝑌 ∈ dom 𝐹)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1002   = wceq 1395  ∃wex 1538   ∈ wcel 2200  {cab 2215  ∀wral 2508  ∃wrex 2509  ∪ cuni 3888  Ord word 4453  suc csuc 4456  dom cdm 4719   ↾ cres 4721  Fun wfun 5312  ⟶wf 5314  ‘cfv 5318  recscrecs 6456

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-recs 6457

This theorem is referenced by:  tfrcl  6516  frecfcllem  6556  frecsuclem  6558