MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uzrdgfni Structured version   Visualization version   GIF version

Theorem uzrdgfni 12574
Description: The recursive definition generator on upper integers is a function. See comment in om2uzrdg 12572. (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 (ℤ𝐶)
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.
StepHypRef Expression
1 uzrdg.3 . . . . . . . . 9 𝑆 = ran 𝑅
21eleq2i 2679 . . . . . . . 8 (𝑧𝑆𝑧 ∈ ran 𝑅)
3 frfnom 7394 . . . . . . . . . 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)
54fneq1i 5885 . . . . . . . . . 10 (𝑅 Fn ω ↔ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω) Fn ω)
63, 5mpbir 219 . . . . . . . . 9 𝑅 Fn ω
7 fvelrnb 6138 . . . . . . . . 9 (𝑅 Fn ω → (𝑧 ∈ ran 𝑅 ↔ ∃𝑤 ∈ ω (𝑅𝑤) = 𝑧))
86, 7ax-mp 5 . . . . . . . 8 (𝑧 ∈ ran 𝑅 ↔ ∃𝑤 ∈ ω (𝑅𝑤) = 𝑧)
92, 8bitri 262 . . . . . . 7 (𝑧𝑆 ↔ ∃𝑤 ∈ ω (𝑅𝑤) = 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
1310, 11, 12, 4om2uzrdg 12572 . . . . . . . . . 10 (𝑤 ∈ ω → (𝑅𝑤) = ⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩)
1410, 11om2uzuzi 12565 . . . . . . . . . . 11 (𝑤 ∈ ω → (𝐺𝑤) ∈ (ℤ𝐶))
15 fvex 6098 . . . . . . . . . . 11 (2nd ‘(𝑅𝑤)) ∈ V
16 opelxpi 5062 . . . . . . . . . . 11 (((𝐺𝑤) ∈ (ℤ𝐶) ∧ (2nd ‘(𝑅𝑤)) ∈ V) → ⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩ ∈ ((ℤ𝐶) × V))
1714, 15, 16sylancl 692 . . . . . . . . . 10 (𝑤 ∈ ω → ⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩ ∈ ((ℤ𝐶) × V))
1813, 17eqeltrd 2687 . . . . . . . . 9 (𝑤 ∈ ω → (𝑅𝑤) ∈ ((ℤ𝐶) × V))
19 eleq1 2675 . . . . . . . . 9 ((𝑅𝑤) = 𝑧 → ((𝑅𝑤) ∈ ((ℤ𝐶) × V) ↔ 𝑧 ∈ ((ℤ𝐶) × V)))
2018, 19syl5ibcom 233 . . . . . . . 8 (𝑤 ∈ ω → ((𝑅𝑤) = 𝑧𝑧 ∈ ((ℤ𝐶) × V)))
2120rexlimiv 3008 . . . . . . 7 (∃𝑤 ∈ ω (𝑅𝑤) = 𝑧𝑧 ∈ ((ℤ𝐶) × V))
229, 21sylbi 205 . . . . . 6 (𝑧𝑆𝑧 ∈ ((ℤ𝐶) × V))
2322ssriv 3571 . . . . 5 𝑆 ⊆ ((ℤ𝐶) × V)
24 xpss 5138 . . . . 5 ((ℤ𝐶) × V) ⊆ (V × V)
2523, 24sstri 3576 . . . 4 𝑆 ⊆ (V × V)
26 df-rel 5035 . . . 4 (Rel 𝑆𝑆 ⊆ (V × V))
2725, 26mpbir 219 . . 3 Rel 𝑆
28 fvex 6098 . . . . . 6 (2nd ‘(𝑅‘(𝐺𝑣))) ∈ V
29 eqeq2 2620 . . . . . . . 8 (𝑤 = (2nd ‘(𝑅‘(𝐺𝑣))) → (𝑧 = 𝑤𝑧 = (2nd ‘(𝑅‘(𝐺𝑣)))))
3029imbi2d 328 . . . . . . 7 (𝑤 = (2nd ‘(𝑅‘(𝐺𝑣))) → ((⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = 𝑤) ↔ (⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = (2nd ‘(𝑅‘(𝐺𝑣))))))
3130albidv 1835 . . . . . 6 (𝑤 = (2nd ‘(𝑅‘(𝐺𝑣))) → (∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = 𝑤) ↔ ∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = (2nd ‘(𝑅‘(𝐺𝑣))))))
3228, 31spcev 3272 . . . . 5 (∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = (2nd ‘(𝑅‘(𝐺𝑣)))) → ∃𝑤𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = 𝑤))
331eleq2i 2679 . . . . . . 7 (⟨𝑣, 𝑧⟩ ∈ 𝑆 ↔ ⟨𝑣, 𝑧⟩ ∈ ran 𝑅)
34 fvelrnb 6138 . . . . . . . 8 (𝑅 Fn ω → (⟨𝑣, 𝑧⟩ ∈ ran 𝑅 ↔ ∃𝑤 ∈ ω (𝑅𝑤) = ⟨𝑣, 𝑧⟩))
356, 34ax-mp 5 . . . . . . 7 (⟨𝑣, 𝑧⟩ ∈ ran 𝑅 ↔ ∃𝑤 ∈ ω (𝑅𝑤) = ⟨𝑣, 𝑧⟩)
3633, 35bitri 262 . . . . . 6 (⟨𝑣, 𝑧⟩ ∈ 𝑆 ↔ ∃𝑤 ∈ ω (𝑅𝑤) = ⟨𝑣, 𝑧⟩)
3713eqeq1d 2611 . . . . . . . . . . . 12 (𝑤 ∈ ω → ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ ↔ ⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩ = ⟨𝑣, 𝑧⟩))
38 fvex 6098 . . . . . . . . . . . . 13 (𝐺𝑤) ∈ V
3938, 15opth1 4864 . . . . . . . . . . . 12 (⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩ = ⟨𝑣, 𝑧⟩ → (𝐺𝑤) = 𝑣)
4037, 39syl6bi 241 . . . . . . . . . . 11 (𝑤 ∈ ω → ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ → (𝐺𝑤) = 𝑣))
4110, 11om2uzf1oi 12569 . . . . . . . . . . . 12 𝐺:ω–1-1-onto→(ℤ𝐶)
42 f1ocnvfv 6412 . . . . . . . . . . . 12 ((𝐺:ω–1-1-onto→(ℤ𝐶) ∧ 𝑤 ∈ ω) → ((𝐺𝑤) = 𝑣 → (𝐺𝑣) = 𝑤))
4341, 42mpan 701 . . . . . . . . . . 11 (𝑤 ∈ ω → ((𝐺𝑤) = 𝑣 → (𝐺𝑣) = 𝑤))
4440, 43syld 45 . . . . . . . . . 10 (𝑤 ∈ ω → ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ → (𝐺𝑣) = 𝑤))
45 fveq2 6088 . . . . . . . . . . 11 ((𝐺𝑣) = 𝑤 → (𝑅‘(𝐺𝑣)) = (𝑅𝑤))
4645fveq2d 6092 . . . . . . . . . 10 ((𝐺𝑣) = 𝑤 → (2nd ‘(𝑅‘(𝐺𝑣))) = (2nd ‘(𝑅𝑤)))
4744, 46syl6 34 . . . . . . . . 9 (𝑤 ∈ ω → ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ → (2nd ‘(𝑅‘(𝐺𝑣))) = (2nd ‘(𝑅𝑤))))
4847imp 443 . . . . . . . 8 ((𝑤 ∈ ω ∧ (𝑅𝑤) = ⟨𝑣, 𝑧⟩) → (2nd ‘(𝑅‘(𝐺𝑣))) = (2nd ‘(𝑅𝑤)))
49 vex 3175 . . . . . . . . . 10 𝑣 ∈ V
50 vex 3175 . . . . . . . . . 10 𝑧 ∈ V
5149, 50op2ndd 7047 . . . . . . . . 9 ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ → (2nd ‘(𝑅𝑤)) = 𝑧)
5251adantl 480 . . . . . . . 8 ((𝑤 ∈ ω ∧ (𝑅𝑤) = ⟨𝑣, 𝑧⟩) → (2nd ‘(𝑅𝑤)) = 𝑧)
5348, 52eqtr2d 2644 . . . . . . 7 ((𝑤 ∈ ω ∧ (𝑅𝑤) = ⟨𝑣, 𝑧⟩) → 𝑧 = (2nd ‘(𝑅‘(𝐺𝑣))))
5453rexlimiva 3009 . . . . . 6 (∃𝑤 ∈ ω (𝑅𝑤) = ⟨𝑣, 𝑧⟩ → 𝑧 = (2nd ‘(𝑅‘(𝐺𝑣))))
5536, 54sylbi 205 . . . . 5 (⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = (2nd ‘(𝑅‘(𝐺𝑣))))
5632, 55mpg 1714 . . . 4 𝑤𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = 𝑤)
5756ax-gen 1712 . . 3 𝑣𝑤𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = 𝑤)
58 dffun5 5803 . . 3 (Fun 𝑆 ↔ (Rel 𝑆 ∧ ∀𝑣𝑤𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = 𝑤)))
5927, 57, 58mpbir2an 956 . 2 Fun 𝑆
60 dmss 5232 . . . . 5 (𝑆 ⊆ ((ℤ𝐶) × V) → dom 𝑆 ⊆ dom ((ℤ𝐶) × V))
6123, 60ax-mp 5 . . . 4 dom 𝑆 ⊆ dom ((ℤ𝐶) × V)
62 dmxpss 5470 . . . 4 dom ((ℤ𝐶) × V) ⊆ (ℤ𝐶)
6361, 62sstri 3576 . . 3 dom 𝑆 ⊆ (ℤ𝐶)
6410, 11, 12, 4uzrdglem 12573 . . . . . 6 (𝑣 ∈ (ℤ𝐶) → ⟨𝑣, (2nd ‘(𝑅‘(𝐺𝑣)))⟩ ∈ ran 𝑅)
6564, 1syl6eleqr 2698 . . . . 5 (𝑣 ∈ (ℤ𝐶) → ⟨𝑣, (2nd ‘(𝑅‘(𝐺𝑣)))⟩ ∈ 𝑆)
6649, 28opeldm 5237 . . . . 5 (⟨𝑣, (2nd ‘(𝑅‘(𝐺𝑣)))⟩ ∈ 𝑆𝑣 ∈ dom 𝑆)
6765, 66syl 17 . . . 4 (𝑣 ∈ (ℤ𝐶) → 𝑣 ∈ dom 𝑆)
6867ssriv 3571 . . 3 (ℤ𝐶) ⊆ dom 𝑆
6963, 68eqssi 3583 . 2 dom 𝑆 = (ℤ𝐶)
70 df-fn 5793 . 2 (𝑆 Fn (ℤ𝐶) ↔ (Fun 𝑆 ∧ dom 𝑆 = (ℤ𝐶)))
7159, 69, 70mpbir2an 956 1 𝑆 Fn (ℤ𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wal 1472   = wceq 1474  wex 1694  wcel 1976  wrex 2896  Vcvv 3172  wss 3539  cop 4130  cmpt 4637   × cxp 5026  ccnv 5027  dom cdm 5028  ran crn 5029  cres 5030  Rel wrel 5033  Fun wfun 5784   Fn wfn 5785  1-1-ontowf1o 5789  cfv 5790  (class class class)co 6527  cmpt2 6529  ωcom 6934  2nd c2nd 7035  reccrdg 7369  1c1 9793   + caddc 9795  cz 11210  cuz 11519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-n0 11140  df-z 11211  df-uz 11520
This theorem is referenced by:  uzrdg0i  12575  uzrdgsuci  12576  seqfn  12630
  Copyright terms: Public domain W3C validator