Theorem uzrdgfni 13899

Description: The recursive definition generator on upper integers is a function. See comment in om2uzrdg 13897. (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 2820	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑆 ↔ 𝑧 ∈ ran 𝑅)
3		frfnom 8380	. . . . . . . . . 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 6597	. . . . . . . . . 10 ⊢ (𝑅 Fn ω ↔ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω) Fn ω)
6	3, 5	mpbir 231	. . . . . . . . 9 ⊢ 𝑅 Fn ω
7		fvelrnb 6903	. . . . . . . . 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 275	. . . . . . 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 13897	. . . . . . . . . 10 ⊢ (𝑤 ∈ ω → (𝑅‘𝑤) = ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩)
14	10, 11	om2uzuzi 13890	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ω → (𝐺‘𝑤) ∈ (ℤ≥‘𝐶))
15		fvex 6853	. . . . . . . . . . 11 ⊢ (2nd ‘(𝑅‘𝑤)) ∈ V
16		opelxpi 5668	. . . . . . . . . . 11 ⊢ (((𝐺‘𝑤) ∈ (ℤ≥‘𝐶) ∧ (2nd ‘(𝑅‘𝑤)) ∈ V) → ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ ∈ ((ℤ≥‘𝐶) × V))
17	14, 15, 16	sylancl 586	. . . . . . . . . 10 ⊢ (𝑤 ∈ ω → ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ ∈ ((ℤ≥‘𝐶) × V))
18	13, 17	eqeltrd 2828	. . . . . . . . 9 ⊢ (𝑤 ∈ ω → (𝑅‘𝑤) ∈ ((ℤ≥‘𝐶) × V))
19		eleq1 2816	. . . . . . . . 9 ⊢ ((𝑅‘𝑤) = 𝑧 → ((𝑅‘𝑤) ∈ ((ℤ≥‘𝐶) × V) ↔ 𝑧 ∈ ((ℤ≥‘𝐶) × V)))
20	18, 19	syl5ibcom 245	. . . . . . . 8 ⊢ (𝑤 ∈ ω → ((𝑅‘𝑤) = 𝑧 → 𝑧 ∈ ((ℤ≥‘𝐶) × V)))
21	20	rexlimiv 3127	. . . . . . 7 ⊢ (∃𝑤 ∈ ω (𝑅‘𝑤) = 𝑧 → 𝑧 ∈ ((ℤ≥‘𝐶) × V))
22	9, 21	sylbi 217	. . . . . 6 ⊢ (𝑧 ∈ 𝑆 → 𝑧 ∈ ((ℤ≥‘𝐶) × V))
23	22	ssriv 3947	. . . . 5 ⊢ 𝑆 ⊆ ((ℤ≥‘𝐶) × V)
24		xpss 5647	. . . . 5 ⊢ ((ℤ≥‘𝐶) × V) ⊆ (V × V)
25	23, 24	sstri 3953	. . . 4 ⊢ 𝑆 ⊆ (V × V)
26		df-rel 5638	. . . 4 ⊢ (Rel 𝑆 ↔ 𝑆 ⊆ (V × V))
27	25, 26	mpbir 231	. . 3 ⊢ Rel 𝑆
28		fvex 6853	. . . . . 6 ⊢ (2nd ‘(𝑅‘(◡𝐺‘𝑣))) ∈ V
29		eqeq2 2741	. . . . . . . 8 ⊢ (𝑤 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))) → (𝑧 = 𝑤 ↔ 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))))
30	29	imbi2d 340	. . . . . . 7 ⊢ (𝑤 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))) → ((⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = 𝑤) ↔ (⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))))))
31	30	albidv 1920	. . . . . 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 3569	. . . . 5 ⊢ (∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))) → ∃𝑤∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = 𝑤))
33	1	eleq2i 2820	. . . . . . 7 ⊢ (⟨𝑣, 𝑧⟩ ∈ 𝑆 ↔ ⟨𝑣, 𝑧⟩ ∈ ran 𝑅)
34		fvelrnb 6903	. . . . . . . 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 275	. . . . . 6 ⊢ (⟨𝑣, 𝑧⟩ ∈ 𝑆 ↔ ∃𝑤 ∈ ω (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩)
37	13	eqeq1d 2731	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ω → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ ↔ ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ = ⟨𝑣, 𝑧⟩))
38		fvex 6853	. . . . . . . . . . . . 13 ⊢ (𝐺‘𝑤) ∈ V
39	38, 15	opth1 5430	. . . . . . . . . . . 12 ⊢ (⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ = ⟨𝑣, 𝑧⟩ → (𝐺‘𝑤) = 𝑣)
40	37, 39	biimtrdi 253	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ω → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → (𝐺‘𝑤) = 𝑣))
41	10, 11	om2uzf1oi 13894	. . . . . . . . . . . 12 ⊢ 𝐺:ω–1-1-onto→(ℤ≥‘𝐶)
42		f1ocnvfv 7235	. . . . . . . . . . . 12 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ 𝑤 ∈ ω) → ((𝐺‘𝑤) = 𝑣 → (◡𝐺‘𝑣) = 𝑤))
43	41, 42	mpan 690	. . . . . . . . . . 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 6845	. . . . . . . . . 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 406	. . . . . . . 8 ⊢ ((𝑤 ∈ ω ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩) → (2nd ‘(𝑅‘(◡𝐺‘𝑣))) = (2nd ‘(𝑅‘𝑤)))
48		vex 3448	. . . . . . . . . 10 ⊢ 𝑣 ∈ V
49		vex 3448	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
50	48, 49	op2ndd 7958	. . . . . . . . 9 ⊢ ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → (2nd ‘(𝑅‘𝑤)) = 𝑧)
51	50	adantl 481	. . . . . . . 8 ⊢ ((𝑤 ∈ ω ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩) → (2nd ‘(𝑅‘𝑤)) = 𝑧)
52	47, 51	eqtr2d 2765	. . . . . . 7 ⊢ ((𝑤 ∈ ω ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩) → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))))
53	52	rexlimiva 3126	. . . . . 6 ⊢ (∃𝑤 ∈ ω (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))))
54	36, 53	sylbi 217	. . . . 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 6514	. . 3 ⊢ (Fun 𝑆 ↔ (Rel 𝑆 ∧ ∀𝑣∃𝑤∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = 𝑤)))
58	27, 56, 57	mpbir2an 711	. 2 ⊢ Fun 𝑆
59		dmss 5856	. . . . 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 6132	. . . 4 ⊢ dom ((ℤ≥‘𝐶) × V) ⊆ (ℤ≥‘𝐶)
62	60, 61	sstri 3953	. . 3 ⊢ dom 𝑆 ⊆ (ℤ≥‘𝐶)
63	10, 11, 12, 4	uzrdglem 13898	. . . . . 6 ⊢ (𝑣 ∈ (ℤ≥‘𝐶) → ⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ ∈ ran 𝑅)
64	63, 1	eleqtrrdi 2839	. . . . 5 ⊢ (𝑣 ∈ (ℤ≥‘𝐶) → ⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ ∈ 𝑆)
65	48, 28	opeldm 5861	. . . . 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 3947	. . 3 ⊢ (ℤ≥‘𝐶) ⊆ dom 𝑆
68	62, 67	eqssi 3960	. 2 ⊢ dom 𝑆 = (ℤ≥‘𝐶)
69		df-fn 6502	. 2 ⊢ (𝑆 Fn (ℤ≥‘𝐶) ↔ (Fun 𝑆 ∧ dom 𝑆 = (ℤ≥‘𝐶)))
70	58, 68, 69	mpbir2an 711	1 ⊢ 𝑆 Fn (ℤ≥‘𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃wrex 3053  Vcvv 3444   ⊆ wss 3911  ⟨cop 4591   ↦ cmpt 5183   × cxp 5629  ^◡ccnv 5630  dom cdm 5631  ran crn 5632   ↾ cres 5633  Rel wrel 5636  Fun wfun 6493   Fn wfn 6494  –1-1-onto→wf1o 6498  ‘cfv 6499  (class class class)co 7369   ∈ cmpo 7371  ωcom 7822  2^nd c2nd 7946  reccrdg 8354  1c1 11045   + caddc 11047  ℤcz 12505  ℤ_≥cuz 12769

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-n0 12419  df-z 12506  df-uz 12770

This theorem is referenced by:  uzrdg0i  13900  uzrdgsuci  13901  seqfn  13954