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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 13312. (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 13314 | . . 3 ⊢ 𝑆 Fn (ℤ≥‘𝐶) |
7 | fnfun 6446 | . . 3 ⊢ (𝑆 Fn (ℤ≥‘𝐶) → Fun 𝑆) | |
8 | 6, 7 | ax-mp 5 | . 2 ⊢ Fun 𝑆 |
9 | 4 | fveq1i 6664 | . . . . 5 ⊢ (𝑅‘∅) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ↾ ω)‘∅) |
10 | opex 5347 | . . . . . 6 ⊢ 〈𝐶, 𝐴〉 ∈ V | |
11 | fr0g 8060 | . . . . . 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 2841 | . . . 4 ⊢ (𝑅‘∅) = 〈𝐶, 𝐴〉 |
14 | frfnom 8059 | . . . . . 6 ⊢ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ↾ ω) Fn ω | |
15 | 4 | fneq1i 6443 | . . . . . 6 ⊢ (𝑅 Fn ω ↔ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ↾ ω) Fn ω) |
16 | 14, 15 | mpbir 232 | . . . . 5 ⊢ 𝑅 Fn ω |
17 | peano1 7590 | . . . . 5 ⊢ ∅ ∈ ω | |
18 | fnfvelrn 6840 | . . . . 5 ⊢ ((𝑅 Fn ω ∧ ∅ ∈ ω) → (𝑅‘∅) ∈ ran 𝑅) | |
19 | 16, 17, 18 | mp2an 688 | . . . 4 ⊢ (𝑅‘∅) ∈ ran 𝑅 |
20 | 13, 19 | eqeltrri 2907 | . . 3 ⊢ 〈𝐶, 𝐴〉 ∈ ran 𝑅 |
21 | 20, 5 | eleqtrri 2909 | . 2 ⊢ 〈𝐶, 𝐴〉 ∈ 𝑆 |
22 | funopfv 6710 | . 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 1528 ∈ wcel 2105 Vcvv 3492 ∅c0 4288 〈cop 4563 ↦ cmpt 5137 ran crn 5549 ↾ cres 5550 Fun wfun 6342 Fn wfn 6343 ‘cfv 6348 (class class class)co 7145 ∈ cmpo 7147 ωcom 7569 reccrdg 8034 1c1 10526 + caddc 10528 ℤcz 11969 ℤ≥cuz 12231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 |
This theorem is referenced by: seq1 13370 |
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