Theorem tfri1 6474

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 1473	. . . 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 6444	. 2 |- ( T. -> F Fn On )
6	5	mptru 1382	1 |- F Fn On

Syntax hints:   /\ wa 104   A.wal 1371   = wceq 1373   T. wtru 1374   e. wcel 2178   _Vcvv 2776   Oncon0 4428   Fun wfun 5284   Fn wfn 5285   ` cfv 5290  recscrecs 6413

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-iord 4431  df-on 4433  df-suc 4436  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-recs 6414

This theorem is referenced by:  tfri2  6475  tfri3  6476