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Mirrors > Home > ILE Home > Th. List > tfri1 | 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" and here is stated as (𝐺‘𝑥) ∈ V. Alternatively ∀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 |
---|---|
tfri1.1 | ⊢ 𝐹 = recs(𝐺) |
tfri1.2 | ⊢ (Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V) |
Ref | Expression |
---|---|
tfri1 | ⊢ 𝐹 Fn On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfri1.1 | . . 3 ⊢ 𝐹 = recs(𝐺) | |
2 | tfri1.2 | . . . . 5 ⊢ (Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V) | |
3 | 2 | ax-gen 1338 | . . . 4 ⊢ ∀𝑥(Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V) |
4 | 3 | a1i 9 | . . 3 ⊢ (⊤ → ∀𝑥(Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V)) |
5 | 1, 4 | tfri1d 5949 | . 2 ⊢ (⊤ → 𝐹 Fn On) |
6 | 5 | trud 1252 | 1 ⊢ 𝐹 Fn On |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ∀wal 1241 = wceq 1243 ⊤wtru 1244 ∈ wcel 1393 Vcvv 2557 Oncon0 4100 Fun wfun 4896 Fn wfn 4897 ‘cfv 4902 recscrecs 5919 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-id 4030 df-iord 4103 df-on 4105 df-suc 4108 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-recs 5920 |
This theorem is referenced by: tfri2 5952 tfri3 5953 |
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