Metamath Proof Explorer

Theorem fin23lem15

Description: Lemma for fin23 . U is a monotone function. (Contributed by Stefan O'Rear, 1-Nov-2014)

Ref Expression
Hypothesis fin23lem.a 
|- U = seqom ( ( i e. _om , u e. _V |-> if ( ( ( t ` i ) i^i u ) = (/) , u , ( ( t ` i ) i^i u ) ) ) , U. ran t )
Assertion fin23lem15 
|- ( ( ( A e. _om /\ B e. _om ) /\ B C_ A ) -> ( U ` A ) C_ ( U ` B ) )

Proof

Step Hyp Ref Expression
1 fin23lem.a 
 |-  U = seqom ( ( i e. _om , u e. _V |-> if ( ( ( t ` i ) i^i u ) = (/) , u , ( ( t ` i ) i^i u ) ) ) , U. ran t )
2 fveq2 
 |-  ( b = B -> ( U ` b ) = ( U ` B ) )
3 2 sseq1d 
 |-  ( b = B -> ( ( U ` b ) C_ ( U ` B ) <-> ( U ` B ) C_ ( U ` B ) ) )
4 fveq2 
 |-  ( b = a -> ( U ` b ) = ( U ` a ) )
5 4 sseq1d 
 |-  ( b = a -> ( ( U ` b ) C_ ( U ` B ) <-> ( U ` a ) C_ ( U ` B ) ) )
6 fveq2 
 |-  ( b = suc a -> ( U ` b ) = ( U ` suc a ) )
7 6 sseq1d 
 |-  ( b = suc a -> ( ( U ` b ) C_ ( U ` B ) <-> ( U ` suc a ) C_ ( U ` B ) ) )
8 fveq2 
 |-  ( b = A -> ( U ` b ) = ( U ` A ) )
9 8 sseq1d 
 |-  ( b = A -> ( ( U ` b ) C_ ( U ` B ) <-> ( U ` A ) C_ ( U ` B ) ) )
10 ssidd 
 |-  ( B e. _om -> ( U ` B ) C_ ( U ` B ) )
11 1 fin23lem13 
 |-  ( a e. _om -> ( U ` suc a ) C_ ( U ` a ) )
12 11 ad2antrr 
 |-  ( ( ( a e. _om /\ B e. _om ) /\ B C_ a ) -> ( U ` suc a ) C_ ( U ` a ) )
13 sstr2 
 |-  ( ( U ` suc a ) C_ ( U ` a ) -> ( ( U ` a ) C_ ( U ` B ) -> ( U ` suc a ) C_ ( U ` B ) ) )
14 12 13 syl 
 |-  ( ( ( a e. _om /\ B e. _om ) /\ B C_ a ) -> ( ( U ` a ) C_ ( U ` B ) -> ( U ` suc a ) C_ ( U ` B ) ) )
15 3 5 7 9 10 14 findsg 
 |-  ( ( ( A e. _om /\ B e. _om ) /\ B C_ A ) -> ( U ` A ) C_ ( U ` B ) )