Theorem orbitclmpt

Description: Version of orbitcl using maps-to notation. (Contributed by Eric Schmidt, 6-Nov-2025)

	Ref	Expression
Hypotheses	orbitclmpt.1	|- F/_ x B
	orbitclmpt.2	|- F/_ x D
	orbitclmpt.3	|- Z = ( rec ( ( x e. _V |-> C ) , A ) " _om )
	orbitclmpt.4	|- ( x = B -> C = D )
Assertion	orbitclmpt	|- ( ( B e. Z /\ D e. V ) -> D e. Z )

Proof

Step	Hyp	Ref	Expression
1		orbitclmpt.1	|- F/_ x B
2		orbitclmpt.2	|- F/_ x D
3		orbitclmpt.3	|- Z = ( rec ( ( x e. _V |-> C ) , A ) " _om )
4		orbitclmpt.4	|- ( x = B -> C = D )
5		elex	|- ( B e. Z -> B e. _V )
6		eqid	|- ( x e. _V |-> C ) = ( x e. _V |-> C )
7	1 2 4 6	fvmptf	|- ( ( B e. _V /\ D e. V ) -> ( ( x e. _V |-> C ) ` B ) = D )
8	5 7	sylan	|- ( ( B e. Z /\ D e. V ) -> ( ( x e. _V |-> C ) ` B ) = D )
9		orbitcl	|- ( B e. ( rec ( ( x e. _V |-> C ) , A ) " _om ) -> ( ( x e. _V |-> C ) ` B ) e. ( rec ( ( x e. _V |-> C ) , A ) " _om ) )
10	3	eleq2i	|- ( B e. Z <-> B e. ( rec ( ( x e. _V |-> C ) , A ) " _om ) )
11	3	eleq2i	|- ( ( ( x e. _V |-> C ) ` B ) e. Z <-> ( ( x e. _V |-> C ) ` B ) e. ( rec ( ( x e. _V |-> C ) , A ) " _om ) )
12	9 10 11	3imtr4i	|- ( B e. Z -> ( ( x e. _V |-> C ) ` B ) e. Z )
13	12	adantr	|- ( ( B e. Z /\ D e. V ) -> ( ( x e. _V |-> C ) ` B ) e. Z )
14	8 13	eqeltrrd	|- ( ( B e. Z /\ D e. V ) -> D e. Z )