Theorem uzrdgfni 12619

Description: The recursive definition generator on upper integers is a function. See comment in om2uzrdg 12617. (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 2680	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑆 ↔ 𝑧 ∈ ran 𝑅)
3		frfnom 7417	. . . . . . . . . 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 5899	. . . . . . . . . 10 ⊢ (𝑅 Fn ω ↔ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω) Fn ω)
6	3, 5	mpbir 220	. . . . . . . . 9 ⊢ 𝑅 Fn ω
7		fvelrnb 6153	. . . . . . . . 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 263	. . . . . . 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 12617	. . . . . . . . . 10 ⊢ (𝑤 ∈ ω → (𝑅‘𝑤) = ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩)
14	10, 11	om2uzuzi 12610	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ω → (𝐺‘𝑤) ∈ (ℤ≥‘𝐶))
15		fvex 6113	. . . . . . . . . . 11 ⊢ (2nd ‘(𝑅‘𝑤)) ∈ V
16		opelxpi 5072	. . . . . . . . . . 11 ⊢ (((𝐺‘𝑤) ∈ (ℤ≥‘𝐶) ∧ (2nd ‘(𝑅‘𝑤)) ∈ V) → ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ ∈ ((ℤ≥‘𝐶) × V))
17	14, 15, 16	sylancl 693	. . . . . . . . . 10 ⊢ (𝑤 ∈ ω → ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ ∈ ((ℤ≥‘𝐶) × V))
18	13, 17	eqeltrd 2688	. . . . . . . . 9 ⊢ (𝑤 ∈ ω → (𝑅‘𝑤) ∈ ((ℤ≥‘𝐶) × V))
19		eleq1 2676	. . . . . . . . 9 ⊢ ((𝑅‘𝑤) = 𝑧 → ((𝑅‘𝑤) ∈ ((ℤ≥‘𝐶) × V) ↔ 𝑧 ∈ ((ℤ≥‘𝐶) × V)))
20	18, 19	syl5ibcom 234	. . . . . . . 8 ⊢ (𝑤 ∈ ω → ((𝑅‘𝑤) = 𝑧 → 𝑧 ∈ ((ℤ≥‘𝐶) × V)))
21	20	rexlimiv 3009	. . . . . . 7 ⊢ (∃𝑤 ∈ ω (𝑅‘𝑤) = 𝑧 → 𝑧 ∈ ((ℤ≥‘𝐶) × V))
22	9, 21	sylbi 206	. . . . . 6 ⊢ (𝑧 ∈ 𝑆 → 𝑧 ∈ ((ℤ≥‘𝐶) × V))
23	22	ssriv 3572	. . . . 5 ⊢ 𝑆 ⊆ ((ℤ≥‘𝐶) × V)
24		xpss 5149	. . . . 5 ⊢ ((ℤ≥‘𝐶) × V) ⊆ (V × V)
25	23, 24	sstri 3577	. . . 4 ⊢ 𝑆 ⊆ (V × V)
26		df-rel 5045	. . . 4 ⊢ (Rel 𝑆 ↔ 𝑆 ⊆ (V × V))
27	25, 26	mpbir 220	. . 3 ⊢ Rel 𝑆
28		fvex 6113	. . . . . 6 ⊢ (2nd ‘(𝑅‘(◡𝐺‘𝑣))) ∈ V
29		eqeq2 2621	. . . . . . . 8 ⊢ (𝑤 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))) → (𝑧 = 𝑤 ↔ 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))))
30	29	imbi2d 329	. . . . . . 7 ⊢ (𝑤 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))) → ((⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = 𝑤) ↔ (⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))))))
31	30	albidv 1836	. . . . . 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 3273	. . . . 5 ⊢ (∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣)))) → ∃𝑤∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = 𝑤))
33	1	eleq2i 2680	. . . . . . 7 ⊢ (⟨𝑣, 𝑧⟩ ∈ 𝑆 ↔ ⟨𝑣, 𝑧⟩ ∈ ran 𝑅)
34		fvelrnb 6153	. . . . . . . 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 263	. . . . . 6 ⊢ (⟨𝑣, 𝑧⟩ ∈ 𝑆 ↔ ∃𝑤 ∈ ω (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩)
37	13	eqeq1d 2612	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ω → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ ↔ ⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ = ⟨𝑣, 𝑧⟩))
38		fvex 6113	. . . . . . . . . . . . 13 ⊢ (𝐺‘𝑤) ∈ V
39	38, 15	opth1 4870	. . . . . . . . . . . 12 ⊢ (⟨(𝐺‘𝑤), (2nd ‘(𝑅‘𝑤))⟩ = ⟨𝑣, 𝑧⟩ → (𝐺‘𝑤) = 𝑣)
40	37, 39	syl6bi 242	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ω → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → (𝐺‘𝑤) = 𝑣))
41	10, 11	om2uzf1oi 12614	. . . . . . . . . . . 12 ⊢ 𝐺:ω–1-1-onto→(ℤ≥‘𝐶)
42		f1ocnvfv 6434	. . . . . . . . . . . 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 46	. . . . . . . . . 10 ⊢ (𝑤 ∈ ω → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → (◡𝐺‘𝑣) = 𝑤))
45		fveq2 6103	. . . . . . . . . . 11 ⊢ ((◡𝐺‘𝑣) = 𝑤 → (𝑅‘(◡𝐺‘𝑣)) = (𝑅‘𝑤))
46	45	fveq2d 6107	. . . . . . . . . 10 ⊢ ((◡𝐺‘𝑣) = 𝑤 → (2nd ‘(𝑅‘(◡𝐺‘𝑣))) = (2nd ‘(𝑅‘𝑤)))
47	44, 46	syl6 34	. . . . . . . . 9 ⊢ (𝑤 ∈ ω → ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → (2nd ‘(𝑅‘(◡𝐺‘𝑣))) = (2nd ‘(𝑅‘𝑤))))
48	47	imp 444	. . . . . . . 8 ⊢ ((𝑤 ∈ ω ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩) → (2nd ‘(𝑅‘(◡𝐺‘𝑣))) = (2nd ‘(𝑅‘𝑤)))
49		vex 3176	. . . . . . . . . 10 ⊢ 𝑣 ∈ V
50		vex 3176	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
51	49, 50	op2ndd 7070	. . . . . . . . 9 ⊢ ((𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → (2nd ‘(𝑅‘𝑤)) = 𝑧)
52	51	adantl 481	. . . . . . . 8 ⊢ ((𝑤 ∈ ω ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩) → (2nd ‘(𝑅‘𝑤)) = 𝑧)
53	48, 52	eqtr2d 2645	. . . . . . 7 ⊢ ((𝑤 ∈ ω ∧ (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩) → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))))
54	53	rexlimiva 3010	. . . . . 6 ⊢ (∃𝑤 ∈ ω (𝑅‘𝑤) = ⟨𝑣, 𝑧⟩ → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))))
55	36, 54	sylbi 206	. . . . 5 ⊢ (⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = (2nd ‘(𝑅‘(◡𝐺‘𝑣))))
56	32, 55	mpg 1715	. . . 4 ⊢ ∃𝑤∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = 𝑤)
57	56	ax-gen 1713	. . 3 ⊢ ∀𝑣∃𝑤∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = 𝑤)
58		dffun5 5817	. . 3 ⊢ (Fun 𝑆 ↔ (Rel 𝑆 ∧ ∀𝑣∃𝑤∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆 → 𝑧 = 𝑤)))
59	27, 57, 58	mpbir2an 957	. 2 ⊢ Fun 𝑆
60		dmss 5245	. . . . 5 ⊢ (𝑆 ⊆ ((ℤ≥‘𝐶) × V) → dom 𝑆 ⊆ dom ((ℤ≥‘𝐶) × V))
61	23, 60	ax-mp 5	. . . 4 ⊢ dom 𝑆 ⊆ dom ((ℤ≥‘𝐶) × V)
62		dmxpss 5484	. . . 4 ⊢ dom ((ℤ≥‘𝐶) × V) ⊆ (ℤ≥‘𝐶)
63	61, 62	sstri 3577	. . 3 ⊢ dom 𝑆 ⊆ (ℤ≥‘𝐶)
64	10, 11, 12, 4	uzrdglem 12618	. . . . . 6 ⊢ (𝑣 ∈ (ℤ≥‘𝐶) → ⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ ∈ ran 𝑅)
65	64, 1	syl6eleqr 2699	. . . . 5 ⊢ (𝑣 ∈ (ℤ≥‘𝐶) → ⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ ∈ 𝑆)
66	49, 28	opeldm 5250	. . . . 5 ⊢ (⟨𝑣, (2nd ‘(𝑅‘(◡𝐺‘𝑣)))⟩ ∈ 𝑆 → 𝑣 ∈ dom 𝑆)
67	65, 66	syl 17	. . . 4 ⊢ (𝑣 ∈ (ℤ≥‘𝐶) → 𝑣 ∈ dom 𝑆)
68	67	ssriv 3572	. . 3 ⊢ (ℤ≥‘𝐶) ⊆ dom 𝑆
69	63, 68	eqssi 3584	. 2 ⊢ dom 𝑆 = (ℤ≥‘𝐶)
70		df-fn 5807	. 2 ⊢ (𝑆 Fn (ℤ≥‘𝐶) ↔ (Fun 𝑆 ∧ dom 𝑆 = (ℤ≥‘𝐶)))
71	59, 69, 70	mpbir2an 957	1 ⊢ 𝑆 Fn (ℤ≥‘𝐶)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  ⟨cop 4131   ↦ cmpt 4643   × cxp 5036  ^◡ccnv 5037  dom cdm 5038  ran crn 5039   ↾ cres 5040  Rel wrel 5043  Fun wfun 5798   Fn wfn 5799  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  ωcom 6957  2^nd c2nd 7058  reccrdg 7392  1c1 9816   + caddc 9818  ℤcz 11254  ℤ_≥cuz 11563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564

This theorem is referenced by:  uzrdg0i  12620  uzrdgsuci  12621  seqfn  12675