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Mirrors > Home > ILE Home > Th. List > frecuzrdg0 | GIF version |
Description: Initial value of a recursive definition generator on upper integers. See comment in frec2uz0d 10013 for the description of 𝐺 as the mapping from ω to (ℤ≥‘𝐶). (Contributed by Jim Kingdon, 27-May-2020.) |
Ref | Expression |
---|---|
frec2uz.1 | ⊢ (𝜑 → 𝐶 ∈ ℤ) |
frec2uz.2 | ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) |
frecuzrdgrrn.a | ⊢ (𝜑 → 𝐴 ∈ 𝑆) |
frecuzrdgrrn.f | ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) |
frecuzrdgrrn.2 | ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) |
frecuzrdgtcl.3 | ⊢ (𝜑 → 𝑇 = ran 𝑅) |
Ref | Expression |
---|---|
frecuzrdg0 | ⊢ (𝜑 → (𝑇‘𝐶) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frec2uz.1 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℤ) | |
2 | frec2uz.2 | . . . 4 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) | |
3 | frecuzrdgrrn.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
4 | frecuzrdgrrn.f | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) | |
5 | frecuzrdgrrn.2 | . . . 4 ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) | |
6 | frecuzrdgtcl.3 | . . . 4 ⊢ (𝜑 → 𝑇 = ran 𝑅) | |
7 | 1, 2, 3, 4, 5, 6 | frecuzrdgtcl 10026 | . . 3 ⊢ (𝜑 → 𝑇:(ℤ≥‘𝐶)⟶𝑆) |
8 | ffun 5211 | . . 3 ⊢ (𝑇:(ℤ≥‘𝐶)⟶𝑆 → Fun 𝑇) | |
9 | 7, 8 | syl 14 | . 2 ⊢ (𝜑 → Fun 𝑇) |
10 | 5 | fveq1i 5354 | . . . . 5 ⊢ (𝑅‘∅) = (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉)‘∅) |
11 | opexg 4088 | . . . . . . 7 ⊢ ((𝐶 ∈ ℤ ∧ 𝐴 ∈ 𝑆) → 〈𝐶, 𝐴〉 ∈ V) | |
12 | 1, 3, 11 | syl2anc 406 | . . . . . 6 ⊢ (𝜑 → 〈𝐶, 𝐴〉 ∈ V) |
13 | frec0g 6224 | . . . . . 6 ⊢ (〈𝐶, 𝐴〉 ∈ V → (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉)‘∅) = 〈𝐶, 𝐴〉) | |
14 | 12, 13 | syl 14 | . . . . 5 ⊢ (𝜑 → (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉)‘∅) = 〈𝐶, 𝐴〉) |
15 | 10, 14 | syl5eq 2144 | . . . 4 ⊢ (𝜑 → (𝑅‘∅) = 〈𝐶, 𝐴〉) |
16 | 1, 2, 3, 4, 5 | frecuzrdgrcl 10024 | . . . . . 6 ⊢ (𝜑 → 𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆)) |
17 | ffn 5208 | . . . . . 6 ⊢ (𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆) → 𝑅 Fn ω) | |
18 | 16, 17 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝑅 Fn ω) |
19 | peano1 4446 | . . . . 5 ⊢ ∅ ∈ ω | |
20 | fnfvelrn 5484 | . . . . 5 ⊢ ((𝑅 Fn ω ∧ ∅ ∈ ω) → (𝑅‘∅) ∈ ran 𝑅) | |
21 | 18, 19, 20 | sylancl 407 | . . . 4 ⊢ (𝜑 → (𝑅‘∅) ∈ ran 𝑅) |
22 | 15, 21 | eqeltrrd 2177 | . . 3 ⊢ (𝜑 → 〈𝐶, 𝐴〉 ∈ ran 𝑅) |
23 | 22, 6 | eleqtrrd 2179 | . 2 ⊢ (𝜑 → 〈𝐶, 𝐴〉 ∈ 𝑇) |
24 | funopfv 5393 | . 2 ⊢ (Fun 𝑇 → (〈𝐶, 𝐴〉 ∈ 𝑇 → (𝑇‘𝐶) = 𝐴)) | |
25 | 9, 23, 24 | sylc 62 | 1 ⊢ (𝜑 → (𝑇‘𝐶) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1299 ∈ wcel 1448 Vcvv 2641 ∅c0 3310 〈cop 3477 ↦ cmpt 3929 ωcom 4442 × cxp 4475 ran crn 4478 Fun wfun 5053 Fn wfn 5054 ⟶wf 5055 ‘cfv 5059 (class class class)co 5706 ∈ cmpo 5708 freccfrec 6217 1c1 7501 + caddc 7503 ℤcz 8906 ℤ≥cuz 9176 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-coll 3983 ax-sep 3986 ax-nul 3994 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-iinf 4440 ax-cnex 7586 ax-resscn 7587 ax-1cn 7588 ax-1re 7589 ax-icn 7590 ax-addcl 7591 ax-addrcl 7592 ax-mulcl 7593 ax-addcom 7595 ax-addass 7597 ax-distr 7599 ax-i2m1 7600 ax-0lt1 7601 ax-0id 7603 ax-rnegex 7604 ax-cnre 7606 ax-pre-ltirr 7607 ax-pre-ltwlin 7608 ax-pre-lttrn 7609 ax-pre-ltadd 7611 |
This theorem depends on definitions: df-bi 116 df-3or 931 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-ral 2380 df-rex 2381 df-reu 2382 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-iun 3762 df-br 3876 df-opab 3930 df-mpt 3931 df-tr 3967 df-id 4153 df-iord 4226 df-on 4228 df-ilim 4229 df-suc 4231 df-iom 4443 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-fv 5067 df-riota 5662 df-ov 5709 df-oprab 5710 df-mpo 5711 df-1st 5969 df-2nd 5970 df-recs 6132 df-frec 6218 df-pnf 7674 df-mnf 7675 df-xr 7676 df-ltxr 7677 df-le 7678 df-sub 7806 df-neg 7807 df-inn 8579 df-n0 8830 df-z 8907 df-uz 9177 |
This theorem is referenced by: (None) |
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