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Metamath Proof Explorer

Theorem wfr3

Description: The principle of Well-Ordered Recursion, part 3 of 3. Finally, we show that F is unique. We do this by showing that any function H with the same properties we proved of F in wfr1 and wfr2 is identical to F . (Contributed by Scott Fenton, 18-Apr-2011) (Revised by Mario Carneiro, 26-Jun-2015) (Revised by Scott Fenton, 18-Nov-2024)

Ref Expression
Hypothesis wfr3.3 
|- F = wrecs ( R , A , G )
Assertion wfr3 
|- ( ( ( R We A /\ R Se A ) /\ ( H Fn A /\ A. z e. A ( H ` z ) = ( G ` ( H |` Pred ( R , A , z ) ) ) ) ) -> F = H )

Proof

Step Hyp Ref Expression
1 wfr3.3 
 |-  F = wrecs ( R , A , G )
2 simpl 
 |-  ( ( ( R We A /\ R Se A ) /\ ( H Fn A /\ A. z e. A ( H ` z ) = ( G ` ( H |` Pred ( R , A , z ) ) ) ) ) -> ( R We A /\ R Se A ) )
3 1 wfr1 
 |-  ( ( R We A /\ R Se A ) -> F Fn A )
4 1 wfr2 
 |-  ( ( ( R We A /\ R Se A ) /\ z e. A ) -> ( F ` z ) = ( G ` ( F |` Pred ( R , A , z ) ) ) )
5 4 ralrimiva 
 |-  ( ( R We A /\ R Se A ) -> A. z e. A ( F ` z ) = ( G ` ( F |` Pred ( R , A , z ) ) ) )
6 3 5 jca 
 |-  ( ( R We A /\ R Se A ) -> ( F Fn A /\ A. z e. A ( F ` z ) = ( G ` ( F |` Pred ( R , A , z ) ) ) ) )
7 6 adantr 
 |-  ( ( ( R We A /\ R Se A ) /\ ( H Fn A /\ A. z e. A ( H ` z ) = ( G ` ( H |` Pred ( R , A , z ) ) ) ) ) -> ( F Fn A /\ A. z e. A ( F ` z ) = ( G ` ( F |` Pred ( R , A , z ) ) ) ) )
8 simpr 
 |-  ( ( ( R We A /\ R Se A ) /\ ( H Fn A /\ A. z e. A ( H ` z ) = ( G ` ( H |` Pred ( R , A , z ) ) ) ) ) -> ( H Fn A /\ A. z e. A ( H ` z ) = ( G ` ( H |` Pred ( R , A , z ) ) ) ) )
9 wfr3g 
 |-  ( ( ( R We A /\ R Se A ) /\ ( F Fn A /\ A. z e. A ( F ` z ) = ( G ` ( F |` Pred ( R , A , z ) ) ) ) /\ ( H Fn A /\ A. z e. A ( H ` z ) = ( G ` ( H |` Pred ( R , A , z ) ) ) ) ) -> F = H )
10 2 7 8 9 syl3anc 
 |-  ( ( ( R We A /\ R Se A ) /\ ( H Fn A /\ A. z e. A ( H ` z ) = ( G ` ( H |` Pred ( R , A , z ) ) ) ) ) -> F = H )