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Theorem tfri1d 6005
 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 is defined "everywhere", which is stated here as . Alternately, would suffice. Given a function satisfying that condition, we define a class of all "acceptable" functions. The final function we're interested in is the union recs of them. is then said to be defined by transfinite recursion. The purpose of the 3 parts of this theorem is to demonstrate properties of . In this first part we show that is a function whose domain is all ordinal numbers. (Contributed by Jim Kingdon, 4-May-2019.) (Revised by Mario Carneiro, 24-May-2019.)
Hypotheses
Ref Expression
tfri1d.1 recs
tfri1d.2
Assertion
Ref Expression
tfri1d
Distinct variable group:   ,
Allowed substitution hints:   ()   ()

Proof of Theorem tfri1d
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2083 . . . . . 6
21tfrlem3 5981 . . . . 5
3 tfri1d.2 . . . . 5
42, 3tfrlemi14d 6003 . . . 4 recs
5 eqid 2083 . . . . 5
65tfrlem7 5987 . . . 4 recs
74, 6jctil 305 . . 3 recs recs
8 df-fn 4955 . . 3 recs recs recs
97, 8sylibr 132 . 2 recs
10 tfri1d.1 . . 3 recs
1110fneq1i 5044 . 2 recs
129, 11sylibr 132 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 102  wal 1283   wceq 1285   wcel 1434  cab 2069  wral 2353  wrex 2354  cvv 2610  con0 4146   cdm 4391   cres 4393   wfun 4946   wfn 4947  cfv 4952  recscrecs 5974 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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-iord 4149  df-on 4151  df-suc 4154  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-recs 5975 This theorem is referenced by:  tfri2d  6006  tfri1  6035  rdgifnon  6049  rdgifnon2  6050  frecfnom  6071
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