# Metamath Proof Explorer

## Theorem tfr1

Description: Principle of Transfinite Recursion, part 1 of 3. Theorem 7.41(1) of TakeutiZaring p. 47. We start with an arbitrary class G , normally a function, and 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 NM, 17-Aug-1994) (Revised by Mario Carneiro, 18-Jan-2015)

Ref Expression
Hypothesis tfr.1 𝐹 = recs ( 𝐺 )
Assertion tfr1 𝐹 Fn On

### Proof

Step Hyp Ref Expression
1 tfr.1 𝐹 = recs ( 𝐺 )
2 eqid { 𝑓 ∣ ∃ 𝑥 ∈ On ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓𝑦 ) ) ) } = { 𝑓 ∣ ∃ 𝑥 ∈ On ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓𝑦 ) ) ) }
3 2 tfrlem7 Fun recs ( 𝐺 )
4 2 tfrlem14 dom recs ( 𝐺 ) = On
5 df-fn ( recs ( 𝐺 ) Fn On ↔ ( Fun recs ( 𝐺 ) ∧ dom recs ( 𝐺 ) = On ) )
6 3 4 5 mpbir2an recs ( 𝐺 ) Fn On
7 1 fneq1i ( 𝐹 Fn On ↔ recs ( 𝐺 ) Fn On )
8 6 7 mpbir 𝐹 Fn On