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

Theorem uzrdgfni 13923
Description: The recursive definition generator on upper integers is a function. See comment in om2uzrdg 13921. (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 2826 . . . . . . . 8 (𝑧𝑆𝑧 ∈ ran 𝑅)
3 frfnom 8435 . . . . . . . . . 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 6647 . . . . . . . . . 10 (𝑅 Fn ω ↔ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω) Fn ω)
63, 5mpbir 230 . . . . . . . . 9 𝑅 Fn ω
7 fvelrnb 6953 . . . . . . . . 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 275 . . . . . . 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 13921 . . . . . . . . . 10 (𝑤 ∈ ω → (𝑅𝑤) = ⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩)
1410, 11om2uzuzi 13914 . . . . . . . . . . 11 (𝑤 ∈ ω → (𝐺𝑤) ∈ (ℤ𝐶))
15 fvex 6905 . . . . . . . . . . 11 (2nd ‘(𝑅𝑤)) ∈ V
16 opelxpi 5714 . . . . . . . . . . 11 (((𝐺𝑤) ∈ (ℤ𝐶) ∧ (2nd ‘(𝑅𝑤)) ∈ V) → ⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩ ∈ ((ℤ𝐶) × V))
1714, 15, 16sylancl 587 . . . . . . . . . 10 (𝑤 ∈ ω → ⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩ ∈ ((ℤ𝐶) × V))
1813, 17eqeltrd 2834 . . . . . . . . 9 (𝑤 ∈ ω → (𝑅𝑤) ∈ ((ℤ𝐶) × V))
19 eleq1 2822 . . . . . . . . 9 ((𝑅𝑤) = 𝑧 → ((𝑅𝑤) ∈ ((ℤ𝐶) × V) ↔ 𝑧 ∈ ((ℤ𝐶) × V)))
2018, 19syl5ibcom 244 . . . . . . . 8 (𝑤 ∈ ω → ((𝑅𝑤) = 𝑧𝑧 ∈ ((ℤ𝐶) × V)))
2120rexlimiv 3149 . . . . . . 7 (∃𝑤 ∈ ω (𝑅𝑤) = 𝑧𝑧 ∈ ((ℤ𝐶) × V))
229, 21sylbi 216 . . . . . 6 (𝑧𝑆𝑧 ∈ ((ℤ𝐶) × V))
2322ssriv 3987 . . . . 5 𝑆 ⊆ ((ℤ𝐶) × V)
24 xpss 5693 . . . . 5 ((ℤ𝐶) × V) ⊆ (V × V)
2523, 24sstri 3992 . . . 4 𝑆 ⊆ (V × V)
26 df-rel 5684 . . . 4 (Rel 𝑆𝑆 ⊆ (V × V))
2725, 26mpbir 230 . . 3 Rel 𝑆
28 fvex 6905 . . . . . 6 (2nd ‘(𝑅‘(𝐺𝑣))) ∈ V
29 eqeq2 2745 . . . . . . . 8 (𝑤 = (2nd ‘(𝑅‘(𝐺𝑣))) → (𝑧 = 𝑤𝑧 = (2nd ‘(𝑅‘(𝐺𝑣)))))
3029imbi2d 341 . . . . . . 7 (𝑤 = (2nd ‘(𝑅‘(𝐺𝑣))) → ((⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = 𝑤) ↔ (⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = (2nd ‘(𝑅‘(𝐺𝑣))))))
3130albidv 1924 . . . . . 6 (𝑤 = (2nd ‘(𝑅‘(𝐺𝑣))) → (∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = 𝑤) ↔ ∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = (2nd ‘(𝑅‘(𝐺𝑣))))))
3228, 31spcev 3597 . . . . 5 (∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = (2nd ‘(𝑅‘(𝐺𝑣)))) → ∃𝑤𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = 𝑤))
331eleq2i 2826 . . . . . . 7 (⟨𝑣, 𝑧⟩ ∈ 𝑆 ↔ ⟨𝑣, 𝑧⟩ ∈ ran 𝑅)
34 fvelrnb 6953 . . . . . . . 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 275 . . . . . 6 (⟨𝑣, 𝑧⟩ ∈ 𝑆 ↔ ∃𝑤 ∈ ω (𝑅𝑤) = ⟨𝑣, 𝑧⟩)
3713eqeq1d 2735 . . . . . . . . . . . 12 (𝑤 ∈ ω → ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ ↔ ⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩ = ⟨𝑣, 𝑧⟩))
38 fvex 6905 . . . . . . . . . . . . 13 (𝐺𝑤) ∈ V
3938, 15opth1 5476 . . . . . . . . . . . 12 (⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩ = ⟨𝑣, 𝑧⟩ → (𝐺𝑤) = 𝑣)
4037, 39syl6bi 253 . . . . . . . . . . 11 (𝑤 ∈ ω → ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ → (𝐺𝑤) = 𝑣))
4110, 11om2uzf1oi 13918 . . . . . . . . . . . 12 𝐺:ω–1-1-onto→(ℤ𝐶)
42 f1ocnvfv 7276 . . . . . . . . . . . 12 ((𝐺:ω–1-1-onto→(ℤ𝐶) ∧ 𝑤 ∈ ω) → ((𝐺𝑤) = 𝑣 → (𝐺𝑣) = 𝑤))
4341, 42mpan 689 . . . . . . . . . . 11 (𝑤 ∈ ω → ((𝐺𝑤) = 𝑣 → (𝐺𝑣) = 𝑤))
4440, 43syld 47 . . . . . . . . . 10 (𝑤 ∈ ω → ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ → (𝐺𝑣) = 𝑤))
45 2fveq3 6897 . . . . . . . . . 10 ((𝐺𝑣) = 𝑤 → (2nd ‘(𝑅‘(𝐺𝑣))) = (2nd ‘(𝑅𝑤)))
4644, 45syl6 35 . . . . . . . . 9 (𝑤 ∈ ω → ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ → (2nd ‘(𝑅‘(𝐺𝑣))) = (2nd ‘(𝑅𝑤))))
4746imp 408 . . . . . . . 8 ((𝑤 ∈ ω ∧ (𝑅𝑤) = ⟨𝑣, 𝑧⟩) → (2nd ‘(𝑅‘(𝐺𝑣))) = (2nd ‘(𝑅𝑤)))
48 vex 3479 . . . . . . . . . 10 𝑣 ∈ V
49 vex 3479 . . . . . . . . . 10 𝑧 ∈ V
5048, 49op2ndd 7986 . . . . . . . . 9 ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ → (2nd ‘(𝑅𝑤)) = 𝑧)
5150adantl 483 . . . . . . . 8 ((𝑤 ∈ ω ∧ (𝑅𝑤) = ⟨𝑣, 𝑧⟩) → (2nd ‘(𝑅𝑤)) = 𝑧)
5247, 51eqtr2d 2774 . . . . . . 7 ((𝑤 ∈ ω ∧ (𝑅𝑤) = ⟨𝑣, 𝑧⟩) → 𝑧 = (2nd ‘(𝑅‘(𝐺𝑣))))
5352rexlimiva 3148 . . . . . 6 (∃𝑤 ∈ ω (𝑅𝑤) = ⟨𝑣, 𝑧⟩ → 𝑧 = (2nd ‘(𝑅‘(𝐺𝑣))))
5436, 53sylbi 216 . . . . 5 (⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = (2nd ‘(𝑅‘(𝐺𝑣))))
5532, 54mpg 1800 . . . 4 𝑤𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = 𝑤)
5655ax-gen 1798 . . 3 𝑣𝑤𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = 𝑤)
57 dffun5 6561 . . 3 (Fun 𝑆 ↔ (Rel 𝑆 ∧ ∀𝑣𝑤𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = 𝑤)))
5827, 56, 57mpbir2an 710 . 2 Fun 𝑆
59 dmss 5903 . . . . 5 (𝑆 ⊆ ((ℤ𝐶) × V) → dom 𝑆 ⊆ dom ((ℤ𝐶) × V))
6023, 59ax-mp 5 . . . 4 dom 𝑆 ⊆ dom ((ℤ𝐶) × V)
61 dmxpss 6171 . . . 4 dom ((ℤ𝐶) × V) ⊆ (ℤ𝐶)
6260, 61sstri 3992 . . 3 dom 𝑆 ⊆ (ℤ𝐶)
6310, 11, 12, 4uzrdglem 13922 . . . . . 6 (𝑣 ∈ (ℤ𝐶) → ⟨𝑣, (2nd ‘(𝑅‘(𝐺𝑣)))⟩ ∈ ran 𝑅)
6463, 1eleqtrrdi 2845 . . . . 5 (𝑣 ∈ (ℤ𝐶) → ⟨𝑣, (2nd ‘(𝑅‘(𝐺𝑣)))⟩ ∈ 𝑆)
6548, 28opeldm 5908 . . . . 5 (⟨𝑣, (2nd ‘(𝑅‘(𝐺𝑣)))⟩ ∈ 𝑆𝑣 ∈ dom 𝑆)
6664, 65syl 17 . . . 4 (𝑣 ∈ (ℤ𝐶) → 𝑣 ∈ dom 𝑆)
6766ssriv 3987 . . 3 (ℤ𝐶) ⊆ dom 𝑆
6862, 67eqssi 3999 . 2 dom 𝑆 = (ℤ𝐶)
69 df-fn 6547 . 2 (𝑆 Fn (ℤ𝐶) ↔ (Fun 𝑆 ∧ dom 𝑆 = (ℤ𝐶)))
7058, 68, 69mpbir2an 710 1 𝑆 Fn (ℤ𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540   = wceq 1542  wex 1782  wcel 2107  wrex 3071  Vcvv 3475  wss 3949  cop 4635  cmpt 5232   × cxp 5675  ccnv 5676  dom cdm 5677  ran crn 5678  cres 5679  Rel wrel 5682  Fun wfun 6538   Fn wfn 6539  1-1-ontowf1o 6543  cfv 6544  (class class class)co 7409  cmpo 7411  ωcom 7855  2nd c2nd 7974  reccrdg 8409  1c1 11111   + caddc 11113  cz 12558  cuz 12822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-n0 12473  df-z 12559  df-uz 12823
This theorem is referenced by:  uzrdg0i  13924  uzrdgsuci  13925  seqfn  13978
  Copyright terms: Public domain W3C validator