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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 πΊ is defined "everywhere", which is stated here as (πΊβπ₯) β V. Alternately, βπ₯ β Onβπ(π Fn π₯ β π β dom πΊ) would suffice. Given a function πΊ satisfying that condition, we define a class π΄ of all "acceptable" functions. The final function we're interested in is the union πΉ = recs(πΊ) of them. πΉ is then said to be defined by transfinite recursion. The purpose of the 3 parts of this theorem is to demonstrate properties of πΉ. In this first part we show that πΉ 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 | β’ πΉ = recs(πΊ) |
tfri1.2 | β’ (Fun πΊ β§ (πΊβπ₯) β V) |
Ref | Expression |
---|---|
tfri1 | β’ πΉ Fn On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfri1.1 | . . 3 β’ πΉ = recs(πΊ) | |
2 | tfri1.2 | . . . . 5 β’ (Fun πΊ β§ (πΊβπ₯) β V) | |
3 | 2 | ax-gen 1449 | . . . 4 β’ βπ₯(Fun πΊ β§ (πΊβπ₯) β V) |
4 | 3 | a1i 9 | . . 3 β’ (β€ β βπ₯(Fun πΊ β§ (πΊβπ₯) β V)) |
5 | 1, 4 | tfri1d 6330 | . 2 β’ (β€ β πΉ Fn On) |
6 | 5 | mptru 1362 | 1 β’ πΉ Fn On |
Colors of variables: wff set class |
Syntax hints: β§ wa 104 βwal 1351 = wceq 1353 β€wtru 1354 β wcel 2148 Vcvv 2737 Oncon0 4360 Fun wfun 5206 Fn wfn 5207 βcfv 5212 recscrecs 6299 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4290 df-iord 4363 df-on 4365 df-suc 4368 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-recs 6300 |
This theorem is referenced by: tfri2 6361 tfri3 6362 |
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