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Mirrors > Home > ILE Home > Th. List > tfri1d | GIF version |
Description: Principle of Transfinite
Recursion, part 1 of 3. Theorem 7.41(1) of
[TakeutiZaring] p. 47, with an
additional condition.
The condition is that 𝐺 is defined "everywhere", which is stated here as (𝐺‘𝑥) ∈ V. Alternately, ∀𝑥 ∈ On∀𝑓(𝑓 Fn 𝑥 → 𝑓 ∈ dom 𝐺) would suffice. Given a function 𝐺 satisfying that condition, we define a class 𝐴 of all "acceptable" functions. The final function we're interested in is the union 𝐹 = recs(𝐺) of them. 𝐹 is then said to be defined by transfinite recursion. The purpose of the 3 parts of this theorem is to demonstrate properties of 𝐹. In this first part we show that 𝐹 is a function whose domain is all ordinal numbers. (Contributed by Jim Kingdon, 4-May-2019.) (Revised by Mario Carneiro, 24-May-2019.) |
Ref | Expression |
---|---|
tfri1d.1 | ⊢ 𝐹 = recs(𝐺) |
tfri1d.2 | ⊢ (𝜑 → ∀𝑥(Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V)) |
Ref | Expression |
---|---|
tfri1d | ⊢ (𝜑 → 𝐹 Fn On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2081 | . . . . . 6 ⊢ {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))} = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))} | |
2 | 1 | tfrlem3 5949 | . . . . 5 ⊢ {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} |
3 | tfri1d.2 | . . . . 5 ⊢ (𝜑 → ∀𝑥(Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V)) | |
4 | 2, 3 | tfrlemi14d 5970 | . . . 4 ⊢ (𝜑 → dom recs(𝐺) = On) |
5 | eqid 2081 | . . . . 5 ⊢ {𝑤 ∣ ∃𝑦 ∈ On (𝑤 Fn 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑤‘𝑧) = (𝐺‘(𝑤 ↾ 𝑧)))} = {𝑤 ∣ ∃𝑦 ∈ On (𝑤 Fn 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑤‘𝑧) = (𝐺‘(𝑤 ↾ 𝑧)))} | |
6 | 5 | tfrlem7 5956 | . . . 4 ⊢ Fun recs(𝐺) |
7 | 4, 6 | jctil 305 | . . 3 ⊢ (𝜑 → (Fun recs(𝐺) ∧ dom recs(𝐺) = On)) |
8 | df-fn 4925 | . . 3 ⊢ (recs(𝐺) Fn On ↔ (Fun recs(𝐺) ∧ dom recs(𝐺) = On)) | |
9 | 7, 8 | sylibr 132 | . 2 ⊢ (𝜑 → recs(𝐺) Fn On) |
10 | tfri1d.1 | . . 3 ⊢ 𝐹 = recs(𝐺) | |
11 | 10 | fneq1i 5013 | . 2 ⊢ (𝐹 Fn On ↔ recs(𝐺) Fn On) |
12 | 9, 11 | sylibr 132 | 1 ⊢ (𝜑 → 𝐹 Fn On) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∀wal 1282 = wceq 1284 ∈ wcel 1433 {cab 2067 ∀wral 2348 ∃wrex 2349 Vcvv 2601 Oncon0 4118 dom cdm 4363 ↾ cres 4365 Fun wfun 4916 Fn wfn 4917 ‘cfv 4922 recscrecs 5942 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-recs 5943 |
This theorem is referenced by: tfri2d 5973 tfri1 5974 rdgifnon 5989 rdgifnon2 5990 frecfnom 6009 frecsuclemdm 6011 frecsuclem3 6013 |
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