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Theorem uzrdgfni 12797
Description: The recursive definition generator on upper integers is a function. See comment in om2uzrdg 12795. (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 2722 . . . . . . . 8 (𝑧𝑆𝑧 ∈ ran 𝑅)
3 frfnom 7575 . . . . . . . . . 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 6023 . . . . . . . . . 10 (𝑅 Fn ω ↔ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω) Fn ω)
63, 5mpbir 221 . . . . . . . . 9 𝑅 Fn ω
7 fvelrnb 6282 . . . . . . . . 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 264 . . . . . . 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 12795 . . . . . . . . . 10 (𝑤 ∈ ω → (𝑅𝑤) = ⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩)
1410, 11om2uzuzi 12788 . . . . . . . . . . 11 (𝑤 ∈ ω → (𝐺𝑤) ∈ (ℤ𝐶))
15 fvex 6239 . . . . . . . . . . 11 (2nd ‘(𝑅𝑤)) ∈ V
16 opelxpi 5182 . . . . . . . . . . 11 (((𝐺𝑤) ∈ (ℤ𝐶) ∧ (2nd ‘(𝑅𝑤)) ∈ V) → ⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩ ∈ ((ℤ𝐶) × V))
1714, 15, 16sylancl 695 . . . . . . . . . 10 (𝑤 ∈ ω → ⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩ ∈ ((ℤ𝐶) × V))
1813, 17eqeltrd 2730 . . . . . . . . 9 (𝑤 ∈ ω → (𝑅𝑤) ∈ ((ℤ𝐶) × V))
19 eleq1 2718 . . . . . . . . 9 ((𝑅𝑤) = 𝑧 → ((𝑅𝑤) ∈ ((ℤ𝐶) × V) ↔ 𝑧 ∈ ((ℤ𝐶) × V)))
2018, 19syl5ibcom 235 . . . . . . . 8 (𝑤 ∈ ω → ((𝑅𝑤) = 𝑧𝑧 ∈ ((ℤ𝐶) × V)))
2120rexlimiv 3056 . . . . . . 7 (∃𝑤 ∈ ω (𝑅𝑤) = 𝑧𝑧 ∈ ((ℤ𝐶) × V))
229, 21sylbi 207 . . . . . 6 (𝑧𝑆𝑧 ∈ ((ℤ𝐶) × V))
2322ssriv 3640 . . . . 5 𝑆 ⊆ ((ℤ𝐶) × V)
24 xpss 5159 . . . . 5 ((ℤ𝐶) × V) ⊆ (V × V)
2523, 24sstri 3645 . . . 4 𝑆 ⊆ (V × V)
26 df-rel 5150 . . . 4 (Rel 𝑆𝑆 ⊆ (V × V))
2725, 26mpbir 221 . . 3 Rel 𝑆
28 fvex 6239 . . . . . 6 (2nd ‘(𝑅‘(𝐺𝑣))) ∈ V
29 eqeq2 2662 . . . . . . . 8 (𝑤 = (2nd ‘(𝑅‘(𝐺𝑣))) → (𝑧 = 𝑤𝑧 = (2nd ‘(𝑅‘(𝐺𝑣)))))
3029imbi2d 329 . . . . . . 7 (𝑤 = (2nd ‘(𝑅‘(𝐺𝑣))) → ((⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = 𝑤) ↔ (⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = (2nd ‘(𝑅‘(𝐺𝑣))))))
3130albidv 1889 . . . . . 6 (𝑤 = (2nd ‘(𝑅‘(𝐺𝑣))) → (∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = 𝑤) ↔ ∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = (2nd ‘(𝑅‘(𝐺𝑣))))))
3228, 31spcev 3331 . . . . 5 (∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = (2nd ‘(𝑅‘(𝐺𝑣)))) → ∃𝑤𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = 𝑤))
331eleq2i 2722 . . . . . . 7 (⟨𝑣, 𝑧⟩ ∈ 𝑆 ↔ ⟨𝑣, 𝑧⟩ ∈ ran 𝑅)
34 fvelrnb 6282 . . . . . . . 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 264 . . . . . 6 (⟨𝑣, 𝑧⟩ ∈ 𝑆 ↔ ∃𝑤 ∈ ω (𝑅𝑤) = ⟨𝑣, 𝑧⟩)
3713eqeq1d 2653 . . . . . . . . . . . 12 (𝑤 ∈ ω → ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ ↔ ⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩ = ⟨𝑣, 𝑧⟩))
38 fvex 6239 . . . . . . . . . . . . 13 (𝐺𝑤) ∈ V
3938, 15opth1 4973 . . . . . . . . . . . 12 (⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩ = ⟨𝑣, 𝑧⟩ → (𝐺𝑤) = 𝑣)
4037, 39syl6bi 243 . . . . . . . . . . 11 (𝑤 ∈ ω → ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ → (𝐺𝑤) = 𝑣))
4110, 11om2uzf1oi 12792 . . . . . . . . . . . 12 𝐺:ω–1-1-onto→(ℤ𝐶)
42 f1ocnvfv 6574 . . . . . . . . . . . 12 ((𝐺:ω–1-1-onto→(ℤ𝐶) ∧ 𝑤 ∈ ω) → ((𝐺𝑤) = 𝑣 → (𝐺𝑣) = 𝑤))
4341, 42mpan 706 . . . . . . . . . . 11 (𝑤 ∈ ω → ((𝐺𝑤) = 𝑣 → (𝐺𝑣) = 𝑤))
4440, 43syld 47 . . . . . . . . . 10 (𝑤 ∈ ω → ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ → (𝐺𝑣) = 𝑤))
45 fveq2 6229 . . . . . . . . . . 11 ((𝐺𝑣) = 𝑤 → (𝑅‘(𝐺𝑣)) = (𝑅𝑤))
4645fveq2d 6233 . . . . . . . . . 10 ((𝐺𝑣) = 𝑤 → (2nd ‘(𝑅‘(𝐺𝑣))) = (2nd ‘(𝑅𝑤)))
4744, 46syl6 35 . . . . . . . . 9 (𝑤 ∈ ω → ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ → (2nd ‘(𝑅‘(𝐺𝑣))) = (2nd ‘(𝑅𝑤))))
4847imp 444 . . . . . . . 8 ((𝑤 ∈ ω ∧ (𝑅𝑤) = ⟨𝑣, 𝑧⟩) → (2nd ‘(𝑅‘(𝐺𝑣))) = (2nd ‘(𝑅𝑤)))
49 vex 3234 . . . . . . . . . 10 𝑣 ∈ V
50 vex 3234 . . . . . . . . . 10 𝑧 ∈ V
5149, 50op2ndd 7221 . . . . . . . . 9 ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ → (2nd ‘(𝑅𝑤)) = 𝑧)
5251adantl 481 . . . . . . . 8 ((𝑤 ∈ ω ∧ (𝑅𝑤) = ⟨𝑣, 𝑧⟩) → (2nd ‘(𝑅𝑤)) = 𝑧)
5348, 52eqtr2d 2686 . . . . . . 7 ((𝑤 ∈ ω ∧ (𝑅𝑤) = ⟨𝑣, 𝑧⟩) → 𝑧 = (2nd ‘(𝑅‘(𝐺𝑣))))
5453rexlimiva 3057 . . . . . 6 (∃𝑤 ∈ ω (𝑅𝑤) = ⟨𝑣, 𝑧⟩ → 𝑧 = (2nd ‘(𝑅‘(𝐺𝑣))))
5536, 54sylbi 207 . . . . 5 (⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = (2nd ‘(𝑅‘(𝐺𝑣))))
5632, 55mpg 1764 . . . 4 𝑤𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = 𝑤)
5756ax-gen 1762 . . 3 𝑣𝑤𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = 𝑤)
58 dffun5 5939 . . 3 (Fun 𝑆 ↔ (Rel 𝑆 ∧ ∀𝑣𝑤𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = 𝑤)))
5927, 57, 58mpbir2an 975 . 2 Fun 𝑆
60 dmss 5355 . . . . 5 (𝑆 ⊆ ((ℤ𝐶) × V) → dom 𝑆 ⊆ dom ((ℤ𝐶) × V))
6123, 60ax-mp 5 . . . 4 dom 𝑆 ⊆ dom ((ℤ𝐶) × V)
62 dmxpss 5600 . . . 4 dom ((ℤ𝐶) × V) ⊆ (ℤ𝐶)
6361, 62sstri 3645 . . 3 dom 𝑆 ⊆ (ℤ𝐶)
6410, 11, 12, 4uzrdglem 12796 . . . . . 6 (𝑣 ∈ (ℤ𝐶) → ⟨𝑣, (2nd ‘(𝑅‘(𝐺𝑣)))⟩ ∈ ran 𝑅)
6564, 1syl6eleqr 2741 . . . . 5 (𝑣 ∈ (ℤ𝐶) → ⟨𝑣, (2nd ‘(𝑅‘(𝐺𝑣)))⟩ ∈ 𝑆)
6649, 28opeldm 5360 . . . . 5 (⟨𝑣, (2nd ‘(𝑅‘(𝐺𝑣)))⟩ ∈ 𝑆𝑣 ∈ dom 𝑆)
6765, 66syl 17 . . . 4 (𝑣 ∈ (ℤ𝐶) → 𝑣 ∈ dom 𝑆)
6867ssriv 3640 . . 3 (ℤ𝐶) ⊆ dom 𝑆
6963, 68eqssi 3652 . 2 dom 𝑆 = (ℤ𝐶)
70 df-fn 5929 . 2 (𝑆 Fn (ℤ𝐶) ↔ (Fun 𝑆 ∧ dom 𝑆 = (ℤ𝐶)))
7159, 69, 70mpbir2an 975 1 𝑆 Fn (ℤ𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1521   = wceq 1523  wex 1744  wcel 2030  wrex 2942  Vcvv 3231  wss 3607  cop 4216  cmpt 4762   × cxp 5141  ccnv 5142  dom cdm 5143  ran crn 5144  cres 5145  Rel wrel 5148  Fun wfun 5920   Fn wfn 5921  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690  cmpt2 6692  ωcom 7107  2nd c2nd 7209  reccrdg 7550  1c1 9975   + caddc 9977  cz 11415  cuz 11725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726
This theorem is referenced by:  uzrdg0i  12798  uzrdgsuci  12799  seqfn  12853
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