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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", 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 |
---|---|
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 1442 | . . . 4 ⊢ ∀𝑥(Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V) |
4 | 3 | a1i 9 | . . 3 ⊢ (⊤ → ∀𝑥(Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V)) |
5 | 1, 4 | tfri1d 6311 | . 2 ⊢ (⊤ → 𝐹 Fn On) |
6 | 5 | mptru 1357 | 1 ⊢ 𝐹 Fn On |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ∀wal 1346 = wceq 1348 ⊤wtru 1349 ∈ wcel 2141 Vcvv 2730 Oncon0 4346 Fun wfun 5190 Fn wfn 5191 ‘cfv 5196 recscrecs 6280 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-iord 4349 df-on 4351 df-suc 4354 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-recs 6281 |
This theorem is referenced by: tfri2 6342 tfri3 6343 |
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