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Mirrors > Home > ILE Home > Th. List > frecuzrdgtclt | GIF version |
Description: The recursive definition generator on upper integers is a function. (Contributed by Jim Kingdon, 22-Apr-2022.) |
Ref | Expression |
---|---|
frecuzrdgrclt.c | ⊢ (𝜑 → 𝐶 ∈ ℤ) |
frecuzrdgrclt.a | ⊢ (𝜑 → 𝐴 ∈ 𝑆) |
frecuzrdgrclt.t | ⊢ (𝜑 → 𝑆 ⊆ 𝑇) |
frecuzrdgrclt.f | ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) |
frecuzrdgrclt.r | ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) |
frecuzrdgtclt.3 | ⊢ (𝜑 → 𝑃 = ran 𝑅) |
Ref | Expression |
---|---|
frecuzrdgtclt | ⊢ (𝜑 → 𝑃:(ℤ≥‘𝐶)⟶𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frecuzrdgrclt.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℤ) | |
2 | frecuzrdgrclt.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
3 | frecuzrdgrclt.t | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ 𝑇) | |
4 | frecuzrdgrclt.f | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) | |
5 | frecuzrdgrclt.r | . . . . 5 ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) | |
6 | 1, 2, 3, 4, 5 | frecuzrdgfun 10451 | . . . 4 ⊢ (𝜑 → Fun ran 𝑅) |
7 | frecuzrdgtclt.3 | . . . . 5 ⊢ (𝜑 → 𝑃 = ran 𝑅) | |
8 | 7 | funeqd 5257 | . . . 4 ⊢ (𝜑 → (Fun 𝑃 ↔ Fun ran 𝑅)) |
9 | 6, 8 | mpbird 167 | . . 3 ⊢ (𝜑 → Fun 𝑃) |
10 | 7 | dmeqd 4847 | . . . 4 ⊢ (𝜑 → dom 𝑃 = dom ran 𝑅) |
11 | 1, 2, 3, 4, 5 | frecuzrdgdom 10449 | . . . 4 ⊢ (𝜑 → dom ran 𝑅 = (ℤ≥‘𝐶)) |
12 | 10, 11 | eqtrd 2222 | . . 3 ⊢ (𝜑 → dom 𝑃 = (ℤ≥‘𝐶)) |
13 | df-fn 5238 | . . 3 ⊢ (𝑃 Fn (ℤ≥‘𝐶) ↔ (Fun 𝑃 ∧ dom 𝑃 = (ℤ≥‘𝐶))) | |
14 | 9, 12, 13 | sylanbrc 417 | . 2 ⊢ (𝜑 → 𝑃 Fn (ℤ≥‘𝐶)) |
15 | 1, 2, 3, 4, 5 | frecuzrdgrclt 10446 | . . . 4 ⊢ (𝜑 → 𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆)) |
16 | frn 5393 | . . . 4 ⊢ (𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆) → ran 𝑅 ⊆ ((ℤ≥‘𝐶) × 𝑆)) | |
17 | 15, 16 | syl 14 | . . 3 ⊢ (𝜑 → ran 𝑅 ⊆ ((ℤ≥‘𝐶) × 𝑆)) |
18 | 7, 17 | eqsstrd 3206 | . 2 ⊢ (𝜑 → 𝑃 ⊆ ((ℤ≥‘𝐶) × 𝑆)) |
19 | dff2 5681 | . 2 ⊢ (𝑃:(ℤ≥‘𝐶)⟶𝑆 ↔ (𝑃 Fn (ℤ≥‘𝐶) ∧ 𝑃 ⊆ ((ℤ≥‘𝐶) × 𝑆))) | |
20 | 14, 18, 19 | sylanbrc 417 | 1 ⊢ (𝜑 → 𝑃:(ℤ≥‘𝐶)⟶𝑆) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2160 ⊆ wss 3144 〈cop 3610 ωcom 4607 × cxp 4642 dom cdm 4644 ran crn 4645 Fun wfun 5229 Fn wfn 5230 ⟶wf 5231 ‘cfv 5235 (class class class)co 5896 ∈ cmpo 5898 freccfrec 6415 1c1 7842 + caddc 7844 ℤcz 9283 ℤ≥cuz 9558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 ax-cnex 7932 ax-resscn 7933 ax-1cn 7934 ax-1re 7935 ax-icn 7936 ax-addcl 7937 ax-addrcl 7938 ax-mulcl 7939 ax-addcom 7941 ax-addass 7943 ax-distr 7945 ax-i2m1 7946 ax-0lt1 7947 ax-0id 7949 ax-rnegex 7950 ax-cnre 7952 ax-pre-ltirr 7953 ax-pre-ltwlin 7954 ax-pre-lttrn 7955 ax-pre-ltadd 7957 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-iord 4384 df-on 4386 df-ilim 4387 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-riota 5852 df-ov 5899 df-oprab 5900 df-mpo 5901 df-1st 6165 df-2nd 6166 df-recs 6330 df-frec 6416 df-pnf 8024 df-mnf 8025 df-xr 8026 df-ltxr 8027 df-le 8028 df-sub 8160 df-neg 8161 df-inn 8950 df-n0 9207 df-z 9284 df-uz 9559 |
This theorem is referenced by: frecuzrdg0t 10453 frecuzrdgsuctlem 10454 seqf 10492 seqf2 10495 |
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