Theorem frecuzrdgfunlem 10405

Description: The recursive definition generator on upper integers produces a a function. (Contributed by Jim Kingdon, 24-Apr-2022.)

Hypotheses

Ref	Expression
frecuzrdgrclt.c	⊢ (𝜑 → 𝐶 ∈ ℤ)
frecuzrdgrclt.a	⊢ (𝜑 → 𝐴 ∈ 𝑆)
frecuzrdgrclt.t	⊢ (𝜑 → 𝑆 ⊆ 𝑇)
frecuzrdgrclt.f	⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
frecuzrdgrclt.r	⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)
frecuzrdgfunlem.g	⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)

Assertion

Ref	Expression
frecuzrdgfunlem	⊢ (𝜑 → Fun ran 𝑅)

Distinct variable groups:   𝑥,𝐶,𝑦   𝑥,𝐹,𝑦   𝑥,𝑆,𝑦   𝑥,𝑇,𝑦   𝜑,𝑥,𝑦   𝑥,𝐺,𝑦   𝑥,𝑅,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem frecuzrdgfunlem

Dummy variables 𝑧 𝑤 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frecuzrdgrclt.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℤ)
2		frecuzrdgrclt.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
3		frecuzrdgrclt.t	. . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ 𝑇)
4		frecuzrdgrclt.f	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
5		frecuzrdgrclt.r	. . . . . 6 ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)
6	1, 2, 3, 4, 5	frecuzrdgrclt 10401	. . . . 5 ⊢ (𝜑 → 𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆))
7		frn 5370	. . . . 5 ⊢ (𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆) → ran 𝑅 ⊆ ((ℤ≥‘𝐶) × 𝑆))
8	6, 7	syl 14	. . . 4 ⊢ (𝜑 → ran 𝑅 ⊆ ((ℤ≥‘𝐶) × 𝑆))
9		xpss 4731	. . . 4 ⊢ ((ℤ≥‘𝐶) × 𝑆) ⊆ (V × V)
10	8, 9	sstrdi 3167	. . 3 ⊢ (𝜑 → ran 𝑅 ⊆ (V × V))
11		df-rel 4630	. . 3 ⊢ (Rel ran 𝑅 ↔ ran 𝑅 ⊆ (V × V))
12	10, 11	sylibr 134	. 2 ⊢ (𝜑 → Rel ran 𝑅)
13		frecuzrdgfunlem.g	. . . . . . . . . 10 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
14	1, 13	frec2uzf1od 10392	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:ω–1-1-onto→(ℤ≥‘𝐶))
15		f1ocnvdm 5776	. . . . . . . . 9 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (◡𝐺‘𝑣) ∈ ω)
16	14, 15	sylan 283	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (◡𝐺‘𝑣) ∈ ω)
17	6	ffvelcdmda 5647	. . . . . . . 8 ⊢ ((𝜑 ∧ (◡𝐺‘𝑣) ∈ ω) → (𝑅‘(◡𝐺‘𝑣)) ∈ ((ℤ≥‘𝐶) × 𝑆))
18	16, 17	syldan 282	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (𝑅‘(◡𝐺‘𝑣)) ∈ ((ℤ≥‘𝐶) × 𝑆))
19		xp2nd 6161	. . . . . . 7 ⊢ ((𝑅‘(◡𝐺‘𝑣)) ∈ ((ℤ≥‘𝐶) × 𝑆) → (2nd ‘(𝑅‘(◡𝐺‘𝑣))) ∈ 𝑆)
20	18, 19	syl 14	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → (2nd ‘(𝑅‘(◡𝐺‘𝑣))) ∈ 𝑆)
21		ffn 5361	. . . . . . . . . 10 ⊢ (𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆) → 𝑅 Fn ω)
22		fvelrnb 5559	. . . . . . . . . 10 ⊢ (𝑅 Fn ω → (⟨𝑣, 𝑧⟩ ∈ ran 𝑅 ↔ ∃𝑤 ∈ ω (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩))
23	6, 21, 22	3syl 17	. . . . . . . . 9 ⊢ (𝜑 → (⟨𝑣, 𝑧⟩ ∈ ran 𝑅 ↔ ∃𝑤 ∈ ω (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩))
24	6	ffvelcdmda 5647	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → (𝑅‘𝑤) ∈ ((ℤ≥‘𝐶) × 𝑆))
25		1st2nd2 6170	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅‘𝑤) ∈ ((ℤ≥‘𝐶) × 𝑆) → (𝑅‘𝑤) = ⟨(1st ‘(𝑅‘𝑤)), (2nd ‘(𝑅‘𝑤))⟩)
26	24, 25	syl 14	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → (𝑅‘𝑤) = ⟨(1st ‘(𝑅‘𝑤)), (2nd ‘(𝑅‘𝑤))⟩)
27	1	adantr 276	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → 𝐶 ∈ ℤ)
28	2	adantr 276	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → 𝐴 ∈ 𝑆)
29	3	adantr 276	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → 𝑆 ⊆ 𝑇)
30	4	adantlr 477	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑤 ∈ ω) ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
31		simpr 110	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → 𝑤 ∈ ω)
32	27, 28, 29, 30, 5, 31, 13	frecuzrdgg 10402	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → (1st ‘(𝑅‘𝑤)) = (𝐺‘𝑤))
33	32	opeq1d 3782	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ⟨(1st ‘(𝑅‘𝑤)), (2nd ‘(𝑅‘𝑤))⟩ = ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩)
34	26, 33	eqtrd 2210	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → (𝑅‘𝑤) = ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩)
35	34	eqeq1d 2186	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ ↔ ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ = ⟨𝑣, 𝑧⟩))
36		vex 2740	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑣 ∈ V
37		vex 2740	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑧 ∈ V
38	36, 37	opth2 4237	. . . . . . . . . . . . . . . . 17 ⊢ (⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ = ⟨𝑣, 𝑧⟩ ↔ ((𝐺‘𝑤) = 𝑣 ∧ (2nd ‘(𝑅‘𝑤)) = 𝑧))
39	38	simplbi 274	. . . . . . . . . . . . . . . 16 ⊢ (⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ = ⟨𝑣, 𝑧⟩ → (𝐺‘𝑤) = 𝑣)
40	35, 39	syl6bi 163	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → (𝐺‘𝑤) = 𝑣))
41		f1ocnvfv 5774	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ 𝑤 ∈ ω) → ((𝐺‘𝑤) = 𝑣 → (◡𝐺‘𝑣) = 𝑤))
42	14, 41	sylan 283	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ((𝐺‘𝑤) = 𝑣 → (◡𝐺‘𝑣) = 𝑤))
43	40, 42	syld 45	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → (◡𝐺‘𝑣) = 𝑤))
44		fveq2 5511	. . . . . . . . . . . . . . 15 ⊢ ((◡𝐺‘𝑣) = 𝑤 → (𝑅‘(◡𝐺‘𝑣)) = (𝑅‘𝑤))
45	44	fveq2d 5515	. . . . . . . . . . . . . 14 ⊢ ((◡𝐺‘𝑣) = 𝑤 → (2nd ‘(𝑅‘(◡𝐺‘𝑣))) = (2nd ‘(𝑅‘𝑤)))
46	43, 45	syl6 33	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → (2nd ‘(𝑅‘(◡𝐺‘𝑣))) = (2nd ‘(𝑅‘𝑤))))
47	46	imp 124	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ ω) ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩) → (2nd ‘(𝑅‘(◡𝐺‘𝑣))) = (2nd ‘(𝑅‘𝑤)))
48	36, 37	op2ndd 6144	. . . . . . . . . . . . 13 ⊢ ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → (2nd ‘(𝑅‘𝑤)) = 𝑧)
49	48	adantl 277	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ ω) ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩) → (2nd ‘(𝑅‘𝑤)) = 𝑧)
50	47, 49	eqtr2d 2211	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ ω) ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩) → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))))
51	50	ex 115	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ ω) → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))))
52	51	rexlimdva 2594	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑤 ∈ ω (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))))
53	23, 52	sylbid 150	. . . . . . . 8 ⊢ (𝜑 → (⟨𝑣, 𝑧⟩ ∈ ran 𝑅 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))))
54	53	alrimiv 1874	. . . . . . 7 ⊢ (𝜑 → ∀𝑧(⟨𝑣, 𝑧⟩ ∈ ran 𝑅 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))))
55	54	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → ∀𝑧(⟨𝑣, 𝑧⟩ ∈ ran 𝑅 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))))
56		eqeq2 2187	. . . . . . . . 9 ⊢ (𝑤 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))) → (𝑧 = 𝑤 ↔ 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))))
57	56	imbi2d 230	. . . . . . . 8 ⊢ (𝑤 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))) → ((⟨𝑣, 𝑧⟩ ∈ ran 𝑅 → 𝑧 = 𝑤) ↔ (⟨𝑣, 𝑧⟩ ∈ ran 𝑅 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))))))
58	57	albidv 1824	. . . . . . 7 ⊢ (𝑤 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))) → (∀𝑧(⟨𝑣, 𝑧⟩ ∈ ran 𝑅 → 𝑧 = 𝑤) ↔ ∀𝑧(⟨𝑣, 𝑧⟩ ∈ ran 𝑅 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))))))
59	58	spcegv 2825	. . . . . 6 ⊢ ((2nd ‘(𝑅‘(◡𝐺‘𝑣))) ∈ 𝑆 → (∀𝑧(⟨𝑣, 𝑧⟩ ∈ ran 𝑅 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))) → ∃𝑤∀𝑧(⟨𝑣, 𝑧⟩ ∈ ran 𝑅 → 𝑧 = 𝑤)))
60	20, 55, 59	sylc 62	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → ∃𝑤∀𝑧(⟨𝑣, 𝑧⟩ ∈ ran 𝑅 → 𝑧 = 𝑤))
61		nfv 1528	. . . . . 6 ⊢ Ⅎ𝑤⟨𝑣, 𝑧⟩ ∈ ran 𝑅
62	61	mo2r 2078	. . . . 5 ⊢ (∃𝑤∀𝑧(⟨𝑣, 𝑧⟩ ∈ ran 𝑅 → 𝑧 = 𝑤) → ∃*𝑧⟨𝑣, 𝑧⟩ ∈ ran 𝑅)
63	60, 62	syl 14	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)) → ∃*𝑧⟨𝑣, 𝑧⟩ ∈ ran 𝑅)
64	1, 2, 3, 4, 5	frecuzrdgdom 10404	. . . . . 6 ⊢ (𝜑 → dom ran 𝑅 = (ℤ≥‘𝐶))
65	64	eleq2d 2247	. . . . 5 ⊢ (𝜑 → (𝑣 ∈ dom ran 𝑅 ↔ 𝑣 ∈ (ℤ≥‘𝐶)))
66	65	pm5.32i 454	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ dom ran 𝑅) ↔ (𝜑 ∧ 𝑣 ∈ (ℤ≥‘𝐶)))
67		df-br 4001	. . . . 5 ⊢ (𝑣ran 𝑅 𝑧 ↔ ⟨𝑣, 𝑧⟩ ∈ ran 𝑅)
68	67	mobii 2063	. . . 4 ⊢ (∃*𝑧 𝑣ran 𝑅 𝑧 ↔ ∃*𝑧⟨𝑣, 𝑧⟩ ∈ ran 𝑅)
69	63, 66, 68	3imtr4i 201	. . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ dom ran 𝑅) → ∃*𝑧 𝑣ran 𝑅 𝑧)
70	69	ralrimiva 2550	. 2 ⊢ (𝜑 → ∀𝑣 ∈ dom ran 𝑅∃*𝑧 𝑣ran 𝑅 𝑧)
71		dffun7 5239	. 2 ⊢ (Fun ran 𝑅 ↔ (Rel ran 𝑅 ∧ ∀𝑣 ∈ dom ran 𝑅∃*𝑧 𝑣ran 𝑅 𝑧))
72	12, 70, 71	sylanbrc 417	1 ⊢ (𝜑 → Fun ran 𝑅)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1351   = wceq 1353  ∃wex 1492  ∃*wmo 2027   ∈ wcel 2148  ∀wral 2455  ∃wrex 2456  Vcvv 2737   ⊆ wss 3129  ⟨cop 3594   class class class wbr 4000   ↦ cmpt 4061  ωcom 4586   × cxp 4621  ^◡ccnv 4622  dom cdm 4623  ran crn 4624  Rel wrel 4628  Fun wfun 5206   Fn wfn 5207  ⟶wf 5208  –1-1-onto→wf1o 5211  ‘cfv 5212  (class class class)co 5869   ∈ cmpo 5871  1^st c1st 6133  2^nd c2nd 6134  freccfrec 6385  1c1 7803   + caddc 7805  ℤcz 9242  ℤ_≥cuz 9517

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-0id 7910  ax-rnegex 7911  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-ltadd 7918

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-inn 8909  df-n0 9166  df-z 9243  df-uz 9518

This theorem is referenced by:  frecuzrdgfun  10406