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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 13862. (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 13864 | . . 3 ⊢ 𝑆 Fn (ℤ≥‘𝐶) |
7 | fnfun 6603 | . . 3 ⊢ (𝑆 Fn (ℤ≥‘𝐶) → Fun 𝑆) | |
8 | 6, 7 | ax-mp 5 | . 2 ⊢ Fun 𝑆 |
9 | 4 | fveq1i 6844 | . . . . 5 ⊢ (𝑅‘∅) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω)‘∅) |
10 | opex 5422 | . . . . . 6 ⊢ ⟨𝐶, 𝐴⟩ ∈ V | |
11 | fr0g 8383 | . . . . . 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 2765 | . . . 4 ⊢ (𝑅‘∅) = ⟨𝐶, 𝐴⟩ |
14 | frfnom 8382 | . . . . . 6 ⊢ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω) Fn ω | |
15 | 4 | fneq1i 6600 | . . . . . 6 ⊢ (𝑅 Fn ω ↔ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω) Fn ω) |
16 | 14, 15 | mpbir 230 | . . . . 5 ⊢ 𝑅 Fn ω |
17 | peano1 7826 | . . . . 5 ⊢ ∅ ∈ ω | |
18 | fnfvelrn 7032 | . . . . 5 ⊢ ((𝑅 Fn ω ∧ ∅ ∈ ω) → (𝑅‘∅) ∈ ran 𝑅) | |
19 | 16, 17, 18 | mp2an 691 | . . . 4 ⊢ (𝑅‘∅) ∈ ran 𝑅 |
20 | 13, 19 | eqeltrri 2835 | . . 3 ⊢ ⟨𝐶, 𝐴⟩ ∈ ran 𝑅 |
21 | 20, 5 | eleqtrri 2837 | . 2 ⊢ ⟨𝐶, 𝐴⟩ ∈ 𝑆 |
22 | funopfv 6895 | . 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 1542 ∈ wcel 2107 Vcvv 3446 ∅c0 4283 ⟨cop 4593 ↦ cmpt 5189 ran crn 5635 ↾ cres 5636 Fun wfun 6491 Fn wfn 6492 ‘cfv 6497 (class class class)co 7358 ∈ cmpo 7360 ωcom 7803 reccrdg 8356 1c1 11053 + caddc 11055 ℤcz 12500 ℤ≥cuz 12764 |
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 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-nn 12155 df-n0 12415 df-z 12501 df-uz 12765 |
This theorem is referenced by: seq1 13920 |
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