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Metamath Proof Explorer

Theorem smoel2

Description: A strictly monotone ordinal function preserves the membership relation. (Contributed by Mario Carneiro, 12-Mar-2013)

Ref Expression
Assertion smoel2 
|- ( ( ( F Fn A /\ Smo F ) /\ ( B e. A /\ C e. B ) ) -> ( F ` C ) e. ( F ` B ) )

Proof

Step Hyp Ref Expression
1 fndm 
 |-  ( F Fn A -> dom F = A )
2 1 eleq2d 
 |-  ( F Fn A -> ( B e. dom F <-> B e. A ) )
3 2 anbi1d 
 |-  ( F Fn A -> ( ( B e. dom F /\ C e. B ) <-> ( B e. A /\ C e. B ) ) )
4 3 biimprd 
 |-  ( F Fn A -> ( ( B e. A /\ C e. B ) -> ( B e. dom F /\ C e. B ) ) )
5 smoel 
 |-  ( ( Smo F /\ B e. dom F /\ C e. B ) -> ( F ` C ) e. ( F ` B ) )
6 5 3expib 
 |-  ( Smo F -> ( ( B e. dom F /\ C e. B ) -> ( F ` C ) e. ( F ` B ) ) )
7 4 6 sylan9 
 |-  ( ( F Fn A /\ Smo F ) -> ( ( B e. A /\ C e. B ) -> ( F ` C ) e. ( F ` B ) ) )
8 7 imp 
 |-  ( ( ( F Fn A /\ Smo F ) /\ ( B e. A /\ C e. B ) ) -> ( F ` C ) e. ( F ` B ) )