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Theorem uzrdgfni 13676
Description: The recursive definition generator on upper integers is a function. See comment in om2uzrdg 13674. (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 2832 . . . . . . . 8 (𝑧𝑆𝑧 ∈ ran 𝑅)
3 frfnom 8257 . . . . . . . . . 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 6528 . . . . . . . . . 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 6827 . . . . . . . . 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 274 . . . . . . 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 13674 . . . . . . . . . 10 (𝑤 ∈ ω → (𝑅𝑤) = ⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩)
1410, 11om2uzuzi 13667 . . . . . . . . . . 11 (𝑤 ∈ ω → (𝐺𝑤) ∈ (ℤ𝐶))
15 fvex 6784 . . . . . . . . . . 11 (2nd ‘(𝑅𝑤)) ∈ V
16 opelxpi 5627 . . . . . . . . . . 11 (((𝐺𝑤) ∈ (ℤ𝐶) ∧ (2nd ‘(𝑅𝑤)) ∈ V) → ⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩ ∈ ((ℤ𝐶) × V))
1714, 15, 16sylancl 586 . . . . . . . . . 10 (𝑤 ∈ ω → ⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩ ∈ ((ℤ𝐶) × V))
1813, 17eqeltrd 2841 . . . . . . . . 9 (𝑤 ∈ ω → (𝑅𝑤) ∈ ((ℤ𝐶) × V))
19 eleq1 2828 . . . . . . . . 9 ((𝑅𝑤) = 𝑧 → ((𝑅𝑤) ∈ ((ℤ𝐶) × V) ↔ 𝑧 ∈ ((ℤ𝐶) × V)))
2018, 19syl5ibcom 244 . . . . . . . 8 (𝑤 ∈ ω → ((𝑅𝑤) = 𝑧𝑧 ∈ ((ℤ𝐶) × V)))
2120rexlimiv 3211 . . . . . . 7 (∃𝑤 ∈ ω (𝑅𝑤) = 𝑧𝑧 ∈ ((ℤ𝐶) × V))
229, 21sylbi 216 . . . . . 6 (𝑧𝑆𝑧 ∈ ((ℤ𝐶) × V))
2322ssriv 3930 . . . . 5 𝑆 ⊆ ((ℤ𝐶) × V)
24 xpss 5606 . . . . 5 ((ℤ𝐶) × V) ⊆ (V × V)
2523, 24sstri 3935 . . . 4 𝑆 ⊆ (V × V)
26 df-rel 5597 . . . 4 (Rel 𝑆𝑆 ⊆ (V × V))
2725, 26mpbir 230 . . 3 Rel 𝑆
28 fvex 6784 . . . . . 6 (2nd ‘(𝑅‘(𝐺𝑣))) ∈ V
29 eqeq2 2752 . . . . . . . 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 1927 . . . . . 6 (𝑤 = (2nd ‘(𝑅‘(𝐺𝑣))) → (∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = 𝑤) ↔ ∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = (2nd ‘(𝑅‘(𝐺𝑣))))))
3228, 31spcev 3544 . . . . 5 (∀𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = (2nd ‘(𝑅‘(𝐺𝑣)))) → ∃𝑤𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = 𝑤))
331eleq2i 2832 . . . . . . 7 (⟨𝑣, 𝑧⟩ ∈ 𝑆 ↔ ⟨𝑣, 𝑧⟩ ∈ ran 𝑅)
34 fvelrnb 6827 . . . . . . . 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 274 . . . . . 6 (⟨𝑣, 𝑧⟩ ∈ 𝑆 ↔ ∃𝑤 ∈ ω (𝑅𝑤) = ⟨𝑣, 𝑧⟩)
3713eqeq1d 2742 . . . . . . . . . . . 12 (𝑤 ∈ ω → ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ ↔ ⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩ = ⟨𝑣, 𝑧⟩))
38 fvex 6784 . . . . . . . . . . . . 13 (𝐺𝑤) ∈ V
3938, 15opth1 5394 . . . . . . . . . . . 12 (⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩ = ⟨𝑣, 𝑧⟩ → (𝐺𝑤) = 𝑣)
4037, 39syl6bi 252 . . . . . . . . . . 11 (𝑤 ∈ ω → ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ → (𝐺𝑤) = 𝑣))
4110, 11om2uzf1oi 13671 . . . . . . . . . . . 12 𝐺:ω–1-1-onto→(ℤ𝐶)
42 f1ocnvfv 7147 . . . . . . . . . . . 12 ((𝐺:ω–1-1-onto→(ℤ𝐶) ∧ 𝑤 ∈ ω) → ((𝐺𝑤) = 𝑣 → (𝐺𝑣) = 𝑤))
4341, 42mpan 687 . . . . . . . . . . 11 (𝑤 ∈ ω → ((𝐺𝑤) = 𝑣 → (𝐺𝑣) = 𝑤))
4440, 43syld 47 . . . . . . . . . 10 (𝑤 ∈ ω → ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ → (𝐺𝑣) = 𝑤))
45 2fveq3 6776 . . . . . . . . . 10 ((𝐺𝑣) = 𝑤 → (2nd ‘(𝑅‘(𝐺𝑣))) = (2nd ‘(𝑅𝑤)))
4644, 45syl6 35 . . . . . . . . 9 (𝑤 ∈ ω → ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ → (2nd ‘(𝑅‘(𝐺𝑣))) = (2nd ‘(𝑅𝑤))))
4746imp 407 . . . . . . . 8 ((𝑤 ∈ ω ∧ (𝑅𝑤) = ⟨𝑣, 𝑧⟩) → (2nd ‘(𝑅‘(𝐺𝑣))) = (2nd ‘(𝑅𝑤)))
48 vex 3435 . . . . . . . . . 10 𝑣 ∈ V
49 vex 3435 . . . . . . . . . 10 𝑧 ∈ V
5048, 49op2ndd 7835 . . . . . . . . 9 ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ → (2nd ‘(𝑅𝑤)) = 𝑧)
5150adantl 482 . . . . . . . 8 ((𝑤 ∈ ω ∧ (𝑅𝑤) = ⟨𝑣, 𝑧⟩) → (2nd ‘(𝑅𝑤)) = 𝑧)
5247, 51eqtr2d 2781 . . . . . . 7 ((𝑤 ∈ ω ∧ (𝑅𝑤) = ⟨𝑣, 𝑧⟩) → 𝑧 = (2nd ‘(𝑅‘(𝐺𝑣))))
5352rexlimiva 3212 . . . . . 6 (∃𝑤 ∈ ω (𝑅𝑤) = ⟨𝑣, 𝑧⟩ → 𝑧 = (2nd ‘(𝑅‘(𝐺𝑣))))
5436, 53sylbi 216 . . . . 5 (⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = (2nd ‘(𝑅‘(𝐺𝑣))))
5532, 54mpg 1804 . . . 4 𝑤𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = 𝑤)
5655ax-gen 1802 . . 3 𝑣𝑤𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = 𝑤)
57 dffun5 6445 . . 3 (Fun 𝑆 ↔ (Rel 𝑆 ∧ ∀𝑣𝑤𝑧(⟨𝑣, 𝑧⟩ ∈ 𝑆𝑧 = 𝑤)))
5827, 56, 57mpbir2an 708 . 2 Fun 𝑆
59 dmss 5810 . . . . 5 (𝑆 ⊆ ((ℤ𝐶) × V) → dom 𝑆 ⊆ dom ((ℤ𝐶) × V))
6023, 59ax-mp 5 . . . 4 dom 𝑆 ⊆ dom ((ℤ𝐶) × V)
61 dmxpss 6073 . . . 4 dom ((ℤ𝐶) × V) ⊆ (ℤ𝐶)
6260, 61sstri 3935 . . 3 dom 𝑆 ⊆ (ℤ𝐶)
6310, 11, 12, 4uzrdglem 13675 . . . . . 6 (𝑣 ∈ (ℤ𝐶) → ⟨𝑣, (2nd ‘(𝑅‘(𝐺𝑣)))⟩ ∈ ran 𝑅)
6463, 1eleqtrrdi 2852 . . . . 5 (𝑣 ∈ (ℤ𝐶) → ⟨𝑣, (2nd ‘(𝑅‘(𝐺𝑣)))⟩ ∈ 𝑆)
6548, 28opeldm 5815 . . . . 5 (⟨𝑣, (2nd ‘(𝑅‘(𝐺𝑣)))⟩ ∈ 𝑆𝑣 ∈ dom 𝑆)
6664, 65syl 17 . . . 4 (𝑣 ∈ (ℤ𝐶) → 𝑣 ∈ dom 𝑆)
6766ssriv 3930 . . 3 (ℤ𝐶) ⊆ dom 𝑆
6862, 67eqssi 3942 . 2 dom 𝑆 = (ℤ𝐶)
69 df-fn 6435 . 2 (𝑆 Fn (ℤ𝐶) ↔ (Fun 𝑆 ∧ dom 𝑆 = (ℤ𝐶)))
7058, 68, 69mpbir2an 708 1 𝑆 Fn (ℤ𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1540   = wceq 1542  wex 1786  wcel 2110  wrex 3067  Vcvv 3431  wss 3892  cop 4573  cmpt 5162   × cxp 5588  ccnv 5589  dom cdm 5590  ran crn 5591  cres 5592  Rel wrel 5595  Fun wfun 6426   Fn wfn 6427  1-1-ontowf1o 6431  cfv 6432  (class class class)co 7271  cmpo 7273  ωcom 7706  2nd c2nd 7823  reccrdg 8231  1c1 10873   + caddc 10875  cz 12319  cuz 12581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582  ax-cnex 10928  ax-resscn 10929  ax-1cn 10930  ax-icn 10931  ax-addcl 10932  ax-addrcl 10933  ax-mulcl 10934  ax-mulrcl 10935  ax-mulcom 10936  ax-addass 10937  ax-mulass 10938  ax-distr 10939  ax-i2m1 10940  ax-1ne0 10941  ax-1rid 10942  ax-rnegex 10943  ax-rrecex 10944  ax-cnre 10945  ax-pre-lttri 10946  ax-pre-lttrn 10947  ax-pre-ltadd 10948  ax-pre-mulgt0 10949
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-riota 7228  df-ov 7274  df-oprab 7275  df-mpo 7276  df-om 7707  df-2nd 7825  df-frecs 8088  df-wrecs 8119  df-recs 8193  df-rdg 8232  df-er 8481  df-en 8717  df-dom 8718  df-sdom 8719  df-pnf 11012  df-mnf 11013  df-xr 11014  df-ltxr 11015  df-le 11016  df-sub 11207  df-neg 11208  df-nn 11974  df-n0 12234  df-z 12320  df-uz 12582
This theorem is referenced by:  uzrdg0i  13677  uzrdgsuci  13678  seqfn  13731
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