Theorem uzrdgfni 13985

Description: The recursive definition generator on upper integers is a function. See comment in om2uzrdg 13983. (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 2857	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑆 ↔ 𝑧 ∈ ran 𝑅)
3		frfnom 8410	. . . . . . . . . 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 6622	. . . . . . . . . 10 ⊢ (𝑅 Fn ω ↔ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω) Fn ω)
6	3, 5	mpbir 234	. . . . . . . . 9 ⊢ 𝑅 Fn ω
7		fvelrnb 6931	. . . . . . . . 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 278	. . . . . . 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 13983	. . . . . . . . . 10 ⊢ (𝑤 ∈ ω → (𝑅‘𝑤) = ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩)
14	10, 11	om2uzuzi 13976	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ω → (𝐺‘𝑤) ∈ (ℤ≥‘𝐶))
15		fvex 6884	. . . . . . . . . . 11 ⊢ (2nd ‘(𝑅‘𝑤)) ∈ V
16		opelxpi 5689	. . . . . . . . . . 11 ⊢ (((𝐺‘𝑤) ∈ (ℤ≥‘𝐶) ∧ (2nd ‘(𝑅‘𝑤)) ∈ V) → ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ ∈ ((ℤ≥‘𝐶) × V))
17	14, 15, 16	sylancl 597	. . . . . . . . . 10 ⊢ (𝑤 ∈ ω → ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ ∈ ((ℤ≥‘𝐶) × V))
18	13, 17	eqeltrd 2865	. . . . . . . . 9 ⊢ (𝑤 ∈ ω → (𝑅‘𝑤) ∈ ((ℤ≥‘𝐶) × V))
19		eleq1 2853	. . . . . . . . 9 ⊢ ((𝑅‘𝑤) = 𝑧 → ((𝑅‘𝑤) ∈ ((ℤ≥‘𝐶) × V) ↔ 𝑧 ∈ ((ℤ≥‘𝐶) × V)))
20	18, 19	syl5ibcom 248	. . . . . . . 8 ⊢ (𝑤 ∈ ω → ((𝑅‘𝑤) = 𝑧 → 𝑧 ∈ ((ℤ≥‘𝐶) × V)))
21	20	rexlimiv 3159	. . . . . . 7 ⊢ (∃𝑤 ∈ ω (𝑅‘𝑤) = 𝑧 → 𝑧 ∈ ((ℤ≥‘𝐶) × V))
22	9, 21	sylbi 220	. . . . . 6 ⊢ (𝑧 ∈ 𝑆 → 𝑧 ∈ ((ℤ≥‘𝐶) × V))
23	22	ssriv 3943	. . . . 5 ⊢ 𝑆 ⊆ ((ℤ≥‘𝐶) × V)
24		xpss 5668	. . . . 5 ⊢ ((ℤ≥‘𝐶) × V) ⊆ (V × V)
25	23, 24	sstri 3948	. . . 4 ⊢ 𝑆 ⊆ (V × V)
26		df-rel 5659	. . . 4 ⊢ (Rel 𝑆 ↔ 𝑆 ⊆ (V × V))
27	25, 26	mpbir 234	. . 3 ⊢ Rel 𝑆
28		fvex 6884	. . . . . 6 ⊢ (2nd ‘(𝑅‘(◡𝐺‘𝑣))) ∈ V
29		eqeq2 2777	. . . . . . . 8 ⊢ (𝑤 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))) → (𝑧 = 𝑤 ↔ 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))))
30	29	imbi2d 343	. . . . . . 7 ⊢ (𝑤 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))) → ((⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = 𝑤) ↔ (⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))))))
31	30	albidv 1943	. . . . . 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 3568	. . . . 5 ⊢ (∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))) → ∃𝑤∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = 𝑤))
33	1	eleq2i 2857	. . . . . . 7 ⊢ (⟨𝑣, 𝑧⟩ ∈ 𝑆 ↔ ⟨𝑣, 𝑧⟩ ∈ ran 𝑅)
34		fvelrnb 6931	. . . . . . . 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 278	. . . . . 6 ⊢ (⟨𝑣, 𝑧⟩ ∈ 𝑆 ↔ ∃𝑤 ∈ ω (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩)
37	13	eqeq1d 2767	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ω → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ ↔ ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ = ⟨𝑣, 𝑧⟩))
38		fvex 6884	. . . . . . . . . . . . 13 ⊢ (𝐺‘𝑤) ∈ V
39	38, 15	opth1 5448	. . . . . . . . . . . 12 ⊢ (⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ = ⟨𝑣, 𝑧⟩ → (𝐺‘𝑤) = 𝑣)
40	37, 39	biimtrdi 256	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ω → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → (𝐺‘𝑤) = 𝑣))
41	10, 11	om2uzf1oi 13980	. . . . . . . . . . . 12 ⊢ 𝐺:ω–1-1-onto→(ℤ≥‘𝐶)
42		f1ocnvfv 7266	. . . . . . . . . . . 12 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ 𝑤 ∈ ω) → ((𝐺‘𝑤) = 𝑣 → (◡𝐺‘𝑣) = 𝑤))
43	41, 42	mpan 702	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ω → ((𝐺‘𝑤) = 𝑣 → (◡𝐺‘𝑣) = 𝑤))
44	40, 43	syld 48	. . . . . . . . . 10 ⊢ (𝑤 ∈ ω → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → (◡𝐺‘𝑣) = 𝑤))
45		2fveq3 6876	. . . . . . . . . 10 ⊢ ((◡𝐺‘𝑣) = 𝑤 → (2nd ‘(𝑅‘(◡𝐺‘𝑣))) = (2nd ‘(𝑅‘𝑤)))
46	44, 45	syl6 36	. . . . . . . . 9 ⊢ (𝑤 ∈ ω → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → (2nd ‘(𝑅‘(◡𝐺‘𝑣))) = (2nd ‘(𝑅‘𝑤))))
47	46	imp 411	. . . . . . . 8 ⊢ ((𝑤 ∈ ω ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩) → (2nd ‘(𝑅‘(◡𝐺‘𝑣))) = (2nd ‘(𝑅‘𝑤)))
48		vex 3461	. . . . . . . . . 10 ⊢ 𝑣 ∈ V
49		vex 3461	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
50	48, 49	op2ndd 7985	. . . . . . . . 9 ⊢ ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → (2nd ‘(𝑅‘𝑤)) = 𝑧)
51	50	adantl 486	. . . . . . . 8 ⊢ ((𝑤 ∈ ω ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩) → (2nd ‘(𝑅‘𝑤)) = 𝑧)
52	47, 51	eqtr2d 2801	. . . . . . 7 ⊢ ((𝑤 ∈ ω ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩) → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))))
53	52	rexlimiva 3158	. . . . . 6 ⊢ (∃𝑤 ∈ ω (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))))
54	36, 53	sylbi 220	. . . . 5 ⊢ (⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))))
55	32, 54	mpg 1820	. . . 4 ⊢ ∃𝑤∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = 𝑤)
56	55	ax-gen 1818	. . 3 ⊢ ∀𝑣∃𝑤∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = 𝑤)
57		dffun5 6539	. . 3 ⊢ (Fun 𝑆 ↔ (Rel 𝑆 ∧ ∀𝑣∃𝑤∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = 𝑤)))
58	27, 56, 57	mpbir2an 723	. 2 ⊢ Fun 𝑆
59		dmss 5883	. . . . 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 6161	. . . 4 ⊢ dom ((ℤ≥‘𝐶) × V) ⊆ (ℤ≥‘𝐶)
62	60, 61	sstri 3948	. . 3 ⊢ dom 𝑆 ⊆ (ℤ≥‘𝐶)
63	10, 11, 12, 4	uzrdglem 13984	. . . . . 6 ⊢ (𝑣 ∈ (ℤ≥‘𝐶) → ⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ ∈ ran 𝑅)
64	63, 1	eleqtrrdi 2876	. . . . 5 ⊢ (𝑣 ∈ (ℤ≥‘𝐶) → ⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ ∈ 𝑆)
65	48, 28	opeldm 5888	. . . . 5 ⊢ (⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ ∈ 𝑆 → 𝑣 ∈ dom 𝑆)
66	64, 65	syl 18	. . . 4 ⊢ (𝑣 ∈ (ℤ≥‘𝐶) → 𝑣 ∈ dom 𝑆)
67	66	ssriv 3943	. . 3 ⊢ (ℤ≥‘𝐶) ⊆ dom 𝑆
68	62, 67	eqssi 3955	. 2 ⊢ dom 𝑆 = (ℤ≥‘𝐶)
69		df-fn 6528	. 2 ⊢ (𝑆 Fn (ℤ≥‘𝐶) ↔ (Fun 𝑆 ∧ dom 𝑆 = (ℤ≥‘𝐶)))
70	58, 68, 69	mpbir2an 723	1 ⊢ 𝑆 Fn (ℤ≥‘𝐶)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1561   = wceq 1563  ∃wex 1802   ∈ wcel 2145  ∃wrex 3089  Vcvv 3457   ⊆ wss 3907  ⟨cop 4591   ↦ cmpt 5186   × cxp 5650  ^◡ccnv 5651  dom cdm 5652  ran crn 5653   ↾ cres 5654  Rel wrel 5657  Fun wfun 6519   Fn wfn 6520  –1-1-onto→wf1o 6524  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402  ωcom 7850  2^nd c2nd 7973  reccrdg 8384  1c1 11089   + caddc 11091  ℤcz 12582  ℤ_≥cuz 12853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-er 8682  df-en 8932  df-dom 8933  df-sdom 8934  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-nn 12225  df-n0 12496  df-z 12583  df-uz 12854

This theorem is referenced by:  uzrdg0i  13986  uzrdgsuci  13987  seqfn  14040