Theorem uzrdgfni 13927

Description: The recursive definition generator on upper integers is a function. See comment in om2uzrdg 13925. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 4-May-2015.)

Hypotheses

Ref	Expression
om2uz.1	⊢ 𝐶 ∈ ℤ
om2uz.2	⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω)
uzrdg.1	⊢ 𝐴 ∈ V
uzrdg.2	⊢ 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)
uzrdg.3	⊢ 𝑆 = ran 𝑅

Assertion

Ref	Expression
uzrdgfni	⊢ 𝑆 Fn (ℤ≥‘𝐶)

Distinct variable groups:   𝑦,𝐴   𝑥,𝑦,𝐶   𝑦,𝐺   𝑥,𝐹,𝑦

Allowed substitution hints:   𝐴(𝑥)   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝐺(𝑥)

Proof of Theorem uzrdgfni

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

Step	Hyp	Ref	Expression
1		uzrdg.3	. . . . . . . . 9 ⊢ 𝑆 = ran 𝑅
2	1	eleq2i 2823	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑆 ↔ 𝑧 ∈ ran 𝑅)
3		frfnom 8437	. . . . . . . . . 10 ⊢ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω) Fn ω
4		uzrdg.2	. . . . . . . . . . 11 ⊢ 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)
5	4	fneq1i 6645	. . . . . . . . . 10 ⊢ (𝑅 Fn ω ↔ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω) Fn ω)
6	3, 5	mpbir 230	. . . . . . . . 9 ⊢ 𝑅 Fn ω
7		fvelrnb 6951	. . . . . . . . 9 ⊢ (𝑅 Fn ω → (𝑧 ∈ ran 𝑅 ↔ ∃𝑤 ∈ ω (𝑅‘𝑤) = 𝑧))
8	6, 7	ax-mp 5	. . . . . . . 8 ⊢ (𝑧 ∈ ran 𝑅 ↔ ∃𝑤 ∈ ω (𝑅‘𝑤) = 𝑧)
9	2, 8	bitri 274	. . . . . . 7 ⊢ (𝑧 ∈ 𝑆 ↔ ∃𝑤 ∈ ω (𝑅‘𝑤) = 𝑧)
10		om2uz.1	. . . . . . . . . . 11 ⊢ 𝐶 ∈ ℤ
11		om2uz.2	. . . . . . . . . . 11 ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω)
12		uzrdg.1	. . . . . . . . . . 11 ⊢ 𝐴 ∈ V
13	10, 11, 12, 4	om2uzrdg 13925	. . . . . . . . . 10 ⊢ (𝑤 ∈ ω → (𝑅‘𝑤) = ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩)
14	10, 11	om2uzuzi 13918	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ω → (𝐺‘𝑤) ∈ (ℤ≥‘𝐶))
15		fvex 6903	. . . . . . . . . . 11 ⊢ (2nd ‘(𝑅‘𝑤)) ∈ V
16		opelxpi 5712	. . . . . . . . . . 11 ⊢ (((𝐺‘𝑤) ∈ (ℤ≥‘𝐶) ∧ (2nd ‘(𝑅‘𝑤)) ∈ V) → ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ ∈ ((ℤ≥‘𝐶) × V))
17	14, 15, 16	sylancl 584	. . . . . . . . . 10 ⊢ (𝑤 ∈ ω → ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ ∈ ((ℤ≥‘𝐶) × V))
18	13, 17	eqeltrd 2831	. . . . . . . . 9 ⊢ (𝑤 ∈ ω → (𝑅‘𝑤) ∈ ((ℤ≥‘𝐶) × V))
19		eleq1 2819	. . . . . . . . 9 ⊢ ((𝑅‘𝑤) = 𝑧 → ((𝑅‘𝑤) ∈ ((ℤ≥‘𝐶) × V) ↔ 𝑧 ∈ ((ℤ≥‘𝐶) × V)))
20	18, 19	syl5ibcom 244	. . . . . . . 8 ⊢ (𝑤 ∈ ω → ((𝑅‘𝑤) = 𝑧 → 𝑧 ∈ ((ℤ≥‘𝐶) × V)))
21	20	rexlimiv 3146	. . . . . . 7 ⊢ (∃𝑤 ∈ ω (𝑅‘𝑤) = 𝑧 → 𝑧 ∈ ((ℤ≥‘𝐶) × V))
22	9, 21	sylbi 216	. . . . . 6 ⊢ (𝑧 ∈ 𝑆 → 𝑧 ∈ ((ℤ≥‘𝐶) × V))
23	22	ssriv 3985	. . . . 5 ⊢ 𝑆 ⊆ ((ℤ≥‘𝐶) × V)
24		xpss 5691	. . . . 5 ⊢ ((ℤ≥‘𝐶) × V) ⊆ (V × V)
25	23, 24	sstri 3990	. . . 4 ⊢ 𝑆 ⊆ (V × V)
26		df-rel 5682	. . . 4 ⊢ (Rel 𝑆 ↔ 𝑆 ⊆ (V × V))
27	25, 26	mpbir 230	. . 3 ⊢ Rel 𝑆
28		fvex 6903	. . . . . 6 ⊢ (2nd ‘(𝑅‘(◡𝐺‘𝑣))) ∈ V
29		eqeq2 2742	. . . . . . . 8 ⊢ (𝑤 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))) → (𝑧 = 𝑤 ↔ 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))))
30	29	imbi2d 339	. . . . . . 7 ⊢ (𝑤 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))) → ((⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = 𝑤) ↔ (⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))))))
31	30	albidv 1921	. . . . . 6 ⊢ (𝑤 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))) → (∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = 𝑤) ↔ ∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))))))
32	28, 31	spcev 3595	. . . . 5 ⊢ (∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))) → ∃𝑤∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = 𝑤))
33	1	eleq2i 2823	. . . . . . 7 ⊢ (⟨𝑣, 𝑧⟩ ∈ 𝑆 ↔ ⟨𝑣, 𝑧⟩ ∈ ran 𝑅)
34		fvelrnb 6951	. . . . . . . 8 ⊢ (𝑅 Fn ω → (⟨𝑣, 𝑧⟩ ∈ ran 𝑅 ↔ ∃𝑤 ∈ ω (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩))
35	6, 34	ax-mp 5	. . . . . . 7 ⊢ (⟨𝑣, 𝑧⟩ ∈ ran 𝑅 ↔ ∃𝑤 ∈ ω (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩)
36	33, 35	bitri 274	. . . . . 6 ⊢ (⟨𝑣, 𝑧⟩ ∈ 𝑆 ↔ ∃𝑤 ∈ ω (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩)
37	13	eqeq1d 2732	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ω → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ ↔ ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ = ⟨𝑣, 𝑧⟩))
38		fvex 6903	. . . . . . . . . . . . 13 ⊢ (𝐺‘𝑤) ∈ V
39	38, 15	opth1 5474	. . . . . . . . . . . 12 ⊢ (⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ = ⟨𝑣, 𝑧⟩ → (𝐺‘𝑤) = 𝑣)
40	37, 39	syl6bi 252	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ω → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → (𝐺‘𝑤) = 𝑣))
41	10, 11	om2uzf1oi 13922	. . . . . . . . . . . 12 ⊢ 𝐺:ω–1-1-onto→(ℤ≥‘𝐶)
42		f1ocnvfv 7278	. . . . . . . . . . . 12 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ 𝑤 ∈ ω) → ((𝐺‘𝑤) = 𝑣 → (◡𝐺‘𝑣) = 𝑤))
43	41, 42	mpan 686	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ω → ((𝐺‘𝑤) = 𝑣 → (◡𝐺‘𝑣) = 𝑤))
44	40, 43	syld 47	. . . . . . . . . 10 ⊢ (𝑤 ∈ ω → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → (◡𝐺‘𝑣) = 𝑤))
45		2fveq3 6895	. . . . . . . . . 10 ⊢ ((◡𝐺‘𝑣) = 𝑤 → (2nd ‘(𝑅‘(◡𝐺‘𝑣))) = (2nd ‘(𝑅‘𝑤)))
46	44, 45	syl6 35	. . . . . . . . 9 ⊢ (𝑤 ∈ ω → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → (2nd ‘(𝑅‘(◡𝐺‘𝑣))) = (2nd ‘(𝑅‘𝑤))))
47	46	imp 405	. . . . . . . 8 ⊢ ((𝑤 ∈ ω ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩) → (2nd ‘(𝑅‘(◡𝐺‘𝑣))) = (2nd ‘(𝑅‘𝑤)))
48		vex 3476	. . . . . . . . . 10 ⊢ 𝑣 ∈ V
49		vex 3476	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
50	48, 49	op2ndd 7988	. . . . . . . . 9 ⊢ ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → (2nd ‘(𝑅‘𝑤)) = 𝑧)
51	50	adantl 480	. . . . . . . 8 ⊢ ((𝑤 ∈ ω ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩) → (2nd ‘(𝑅‘𝑤)) = 𝑧)
52	47, 51	eqtr2d 2771	. . . . . . 7 ⊢ ((𝑤 ∈ ω ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩) → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))))
53	52	rexlimiva 3145	. . . . . 6 ⊢ (∃𝑤 ∈ ω (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))))
54	36, 53	sylbi 216	. . . . 5 ⊢ (⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))))
55	32, 54	mpg 1797	. . . 4 ⊢ ∃𝑤∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = 𝑤)
56	55	ax-gen 1795	. . 3 ⊢ ∀𝑣∃𝑤∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = 𝑤)
57		dffun5 6559	. . 3 ⊢ (Fun 𝑆 ↔ (Rel 𝑆 ∧ ∀𝑣∃𝑤∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = 𝑤)))
58	27, 56, 57	mpbir2an 707	. 2 ⊢ Fun 𝑆
59		dmss 5901	. . . . 5 ⊢ (𝑆 ⊆ ((ℤ≥‘𝐶) × V) → dom 𝑆 ⊆ dom ((ℤ≥‘𝐶) × V))
60	23, 59	ax-mp 5	. . . 4 ⊢ dom 𝑆 ⊆ dom ((ℤ≥‘𝐶) × V)
61		dmxpss 6169	. . . 4 ⊢ dom ((ℤ≥‘𝐶) × V) ⊆ (ℤ≥‘𝐶)
62	60, 61	sstri 3990	. . 3 ⊢ dom 𝑆 ⊆ (ℤ≥‘𝐶)
63	10, 11, 12, 4	uzrdglem 13926	. . . . . 6 ⊢ (𝑣 ∈ (ℤ≥‘𝐶) → ⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ ∈ ran 𝑅)
64	63, 1	eleqtrrdi 2842	. . . . 5 ⊢ (𝑣 ∈ (ℤ≥‘𝐶) → ⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ ∈ 𝑆)
65	48, 28	opeldm 5906	. . . . 5 ⊢ (⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ ∈ 𝑆 → 𝑣 ∈ dom 𝑆)
66	64, 65	syl 17	. . . 4 ⊢ (𝑣 ∈ (ℤ≥‘𝐶) → 𝑣 ∈ dom 𝑆)
67	66	ssriv 3985	. . 3 ⊢ (ℤ≥‘𝐶) ⊆ dom 𝑆
68	62, 67	eqssi 3997	. 2 ⊢ dom 𝑆 = (ℤ≥‘𝐶)
69		df-fn 6545	. 2 ⊢ (𝑆 Fn (ℤ≥‘𝐶) ↔ (Fun 𝑆 ∧ dom 𝑆 = (ℤ≥‘𝐶)))
70	58, 68, 69	mpbir2an 707	1 ⊢ 𝑆 Fn (ℤ≥‘𝐶)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394  ∀wal 1537   = wceq 1539  ∃wex 1779   ∈ wcel 2104  ∃wrex 3068  Vcvv 3472   ⊆ wss 3947  ⟨cop 4633   ↦ cmpt 5230   × cxp 5673  ^◡ccnv 5674  dom cdm 5675  ran crn 5676   ↾ cres 5677  Rel wrel 5680  Fun wfun 6536   Fn wfn 6537  –1-1-onto→wf1o 6541  ‘cfv 6542  (class class class)co 7411   ∈ cmpo 7413  ωcom 7857  2^nd c2nd 7976  reccrdg 8411  1c1 11113   + caddc 11115  ℤcz 12562  ℤ_≥cuz 12826

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-z 12563  df-uz 12827

This theorem is referenced by:  uzrdg0i  13928  uzrdgsuci  13929  seqfn  13982