Theorem inf3lemc

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 for detailed description. (Contributed by NM, 28-Oct-1996)

	Ref	Expression
Hypotheses	inf3lem.1	|- G = ( y e. _V |-> { w e. x | ( w i^i x ) C_ y } )
	inf3lem.2	|- F = ( rec ( G , (/) ) |` _om )
	inf3lem.3	|- A e. _V
	inf3lem.4	|- B e. _V
Assertion	inf3lemc	|- ( A e. _om -> ( F ` suc A ) = ( G ` ( F ` A ) ) )

Proof

Step	Hyp	Ref	Expression
1		inf3lem.1	|- G = ( y e. _V |-> { w e. x | ( w i^i x ) C_ y } )
2		inf3lem.2	|- F = ( rec ( G , (/) ) |` _om )
3		inf3lem.3	|- A e. _V
4		inf3lem.4	|- B e. _V
5		frsuc	|- ( A e. _om -> ( ( rec ( G , (/) ) |` _om ) ` suc A ) = ( G ` ( ( rec ( G , (/) ) |` _om ) ` A ) ) )
6	2	fveq1i	|- ( F ` suc A ) = ( ( rec ( G , (/) ) |` _om ) ` suc A )
7	2	fveq1i	|- ( F ` A ) = ( ( rec ( G , (/) ) |` _om ) ` A )
8	7	fveq2i	|- ( G ` ( F ` A ) ) = ( G ` ( ( rec ( G , (/) ) |` _om ) ` A ) )
9	5 6 8	3eqtr4g	|- ( A e. _om -> ( F ` suc A ) = ( G ` ( F ` A ) ) )