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Mirrors > Home > MPE Home > Th. List > uzrdg0i | Structured version Visualization version GIF version |
Description: Initial value of a recursive definition generator on upper integers. See comment in om2uzrdg 12755. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 18-Nov-2014.) |
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 |
Ref | Expression |
---|---|
uzrdg0i | ⊢ (𝑆‘𝐶) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | om2uz.1 | . . . 4 ⊢ 𝐶 ∈ ℤ | |
2 | om2uz.2 | . . . 4 ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) | |
3 | uzrdg.1 | . . . 4 ⊢ 𝐴 ∈ V | |
4 | uzrdg.2 | . . . 4 ⊢ 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ↾ ω) | |
5 | uzrdg.3 | . . . 4 ⊢ 𝑆 = ran 𝑅 | |
6 | 1, 2, 3, 4, 5 | uzrdgfni 12757 | . . 3 ⊢ 𝑆 Fn (ℤ≥‘𝐶) |
7 | fnfun 5988 | . . 3 ⊢ (𝑆 Fn (ℤ≥‘𝐶) → Fun 𝑆) | |
8 | 6, 7 | ax-mp 5 | . 2 ⊢ Fun 𝑆 |
9 | 4 | fveq1i 6192 | . . . . 5 ⊢ (𝑅‘∅) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ↾ ω)‘∅) |
10 | opex 4932 | . . . . . 6 ⊢ 〈𝐶, 𝐴〉 ∈ V | |
11 | fr0g 7531 | . . . . . 6 ⊢ (〈𝐶, 𝐴〉 ∈ V → ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ↾ ω)‘∅) = 〈𝐶, 𝐴〉) | |
12 | 10, 11 | ax-mp 5 | . . . . 5 ⊢ ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ↾ ω)‘∅) = 〈𝐶, 𝐴〉 |
13 | 9, 12 | eqtri 2644 | . . . 4 ⊢ (𝑅‘∅) = 〈𝐶, 𝐴〉 |
14 | frfnom 7530 | . . . . . 6 ⊢ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ↾ ω) Fn ω | |
15 | 4 | fneq1i 5985 | . . . . . 6 ⊢ (𝑅 Fn ω ↔ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ↾ ω) Fn ω) |
16 | 14, 15 | mpbir 221 | . . . . 5 ⊢ 𝑅 Fn ω |
17 | peano1 7085 | . . . . 5 ⊢ ∅ ∈ ω | |
18 | fnfvelrn 6356 | . . . . 5 ⊢ ((𝑅 Fn ω ∧ ∅ ∈ ω) → (𝑅‘∅) ∈ ran 𝑅) | |
19 | 16, 17, 18 | mp2an 708 | . . . 4 ⊢ (𝑅‘∅) ∈ ran 𝑅 |
20 | 13, 19 | eqeltrri 2698 | . . 3 ⊢ 〈𝐶, 𝐴〉 ∈ ran 𝑅 |
21 | 20, 5 | eleqtrri 2700 | . 2 ⊢ 〈𝐶, 𝐴〉 ∈ 𝑆 |
22 | funopfv 6235 | . 2 ⊢ (Fun 𝑆 → (〈𝐶, 𝐴〉 ∈ 𝑆 → (𝑆‘𝐶) = 𝐴)) | |
23 | 8, 21, 22 | mp2 9 | 1 ⊢ (𝑆‘𝐶) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∅c0 3915 〈cop 4183 ↦ cmpt 4729 ran crn 5115 ↾ cres 5116 Fun wfun 5882 Fn wfn 5883 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 ωcom 7065 reccrdg 7505 1c1 9937 + caddc 9939 ℤcz 11377 ℤ≥cuz 11687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 |
This theorem is referenced by: seq1 12814 |
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