Theorem tfri1 5974

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 ) e. _V. Alternately, A. x e. On A. f ( f Fn x -> f e. 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.)

Hypotheses

Ref	Expression
tfri1.1	|- F = recs ( G )
tfri1.2	|- ( Fun G /\ ( G ` x ) e. _V )

Assertion

Ref	Expression
tfri1	|- F Fn On

Distinct variable group:   x, G

Allowed substitution hint:   F( x)

Proof of Theorem tfri1

Step	Hyp	Ref	Expression
1		tfri1.1	. . 3 |- F = recs ( G )
2		tfri1.2	. . . . 5 |- ( Fun G /\ ( G ` x ) e. _V )
3	2	ax-gen 1378	. . . 4 |- A. x ( Fun G /\ ( G ` x ) e. _V )
4	3	a1i 9	. . 3 |- ( T. -> A. x ( Fun G /\ ( G ` x ) e. _V ) )
5	1, 4	tfri1d 5972	. 2 |- ( T. -> F Fn On )
6	5	trud 1293	1 |- F Fn On

Syntax hints:   /\ wa 102   A.wal 1282   = wceq 1284   T. wtru 1285   e. wcel 1433   _Vcvv 2601   Oncon0 4118   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:  tfri2  5975  tfri3  5976