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 12617. (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 12619 | . . 3 ⊢ 𝑆 Fn (ℤ≥‘𝐶) |
7 | fnfun 5902 | . . 3 ⊢ (𝑆 Fn (ℤ≥‘𝐶) → Fun 𝑆) | |
8 | 6, 7 | ax-mp 5 | . 2 ⊢ Fun 𝑆 |
9 | 4 | fveq1i 6104 | . . . . 5 ⊢ (𝑅‘∅) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ↾ ω)‘∅) |
10 | opex 4859 | . . . . . 6 ⊢ 〈𝐶, 𝐴〉 ∈ V | |
11 | fr0g 7418 | . . . . . 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 2632 | . . . 4 ⊢ (𝑅‘∅) = 〈𝐶, 𝐴〉 |
14 | frfnom 7417 | . . . . . 6 ⊢ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ↾ ω) Fn ω | |
15 | 4 | fneq1i 5899 | . . . . . 6 ⊢ (𝑅 Fn ω ↔ (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ↾ ω) Fn ω) |
16 | 14, 15 | mpbir 220 | . . . . 5 ⊢ 𝑅 Fn ω |
17 | peano1 6977 | . . . . 5 ⊢ ∅ ∈ ω | |
18 | fnfvelrn 6264 | . . . . 5 ⊢ ((𝑅 Fn ω ∧ ∅ ∈ ω) → (𝑅‘∅) ∈ ran 𝑅) | |
19 | 16, 17, 18 | mp2an 704 | . . . 4 ⊢ (𝑅‘∅) ∈ ran 𝑅 |
20 | 13, 19 | eqeltrri 2685 | . . 3 ⊢ 〈𝐶, 𝐴〉 ∈ ran 𝑅 |
21 | 20, 5 | eleqtrri 2687 | . 2 ⊢ 〈𝐶, 𝐴〉 ∈ 𝑆 |
22 | funopfv 6145 | . 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 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 〈cop 4131 ↦ cmpt 4643 ran crn 5039 ↾ cres 5040 Fun wfun 5798 Fn wfn 5799 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ωcom 6957 reccrdg 7392 1c1 9816 + caddc 9818 ℤcz 11254 ℤ≥cuz 11563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 |
This theorem is referenced by: seq1 12676 |
Copyright terms: Public domain | W3C validator |