![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 13132. (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 13134 | . . 3 ⊢ 𝑆 Fn (ℤ≥‘𝐶) |
7 | fnfun 6280 | . . 3 ⊢ (𝑆 Fn (ℤ≥‘𝐶) → Fun 𝑆) | |
8 | 6, 7 | ax-mp 5 | . 2 ⊢ Fun 𝑆 |
9 | 4 | fveq1i 6494 | . . . . 5 ⊢ (𝑅‘∅) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ↾ ω)‘∅) |
10 | opex 5206 | . . . . . 6 ⊢ 〈𝐶, 𝐴〉 ∈ V | |
11 | fr0g 7868 | . . . . . 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 2796 | . . . 4 ⊢ (𝑅‘∅) = 〈𝐶, 𝐴〉 |
14 | frfnom 7867 | . . . . . 6 ⊢ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ↾ ω) Fn ω | |
15 | 4 | fneq1i 6277 | . . . . . 6 ⊢ (𝑅 Fn ω ↔ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ↾ ω) Fn ω) |
16 | 14, 15 | mpbir 223 | . . . . 5 ⊢ 𝑅 Fn ω |
17 | peano1 7410 | . . . . 5 ⊢ ∅ ∈ ω | |
18 | fnfvelrn 6667 | . . . . 5 ⊢ ((𝑅 Fn ω ∧ ∅ ∈ ω) → (𝑅‘∅) ∈ ran 𝑅) | |
19 | 16, 17, 18 | mp2an 679 | . . . 4 ⊢ (𝑅‘∅) ∈ ran 𝑅 |
20 | 13, 19 | eqeltrri 2857 | . . 3 ⊢ 〈𝐶, 𝐴〉 ∈ ran 𝑅 |
21 | 20, 5 | eleqtrri 2859 | . 2 ⊢ 〈𝐶, 𝐴〉 ∈ 𝑆 |
22 | funopfv 6541 | . 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 1507 ∈ wcel 2048 Vcvv 3409 ∅c0 4173 〈cop 4441 ↦ cmpt 5002 ran crn 5401 ↾ cres 5402 Fun wfun 6176 Fn wfn 6177 ‘cfv 6182 (class class class)co 6970 ∈ cmpo 6972 ωcom 7390 reccrdg 7842 1c1 10328 + caddc 10330 ℤcz 11786 ℤ≥cuz 12051 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-2nd 7495 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-er 8081 df-en 8299 df-dom 8300 df-sdom 8301 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-nn 11432 df-n0 11701 df-z 11787 df-uz 12052 |
This theorem is referenced by: seq1 13190 |
Copyright terms: Public domain | W3C validator |