Metamath Proof Explorer


Theorem madecut

Description: Given a section that is a subset of an old set, the cut is a member of the made set. (Contributed by Scott Fenton, 7-Aug-2024)

Ref Expression
Assertion madecut
|- ( ( ( A e. On /\ L < ( L |s R ) e. ( _M ` A ) )

Proof

Step Hyp Ref Expression
1 simplr
 |-  ( ( ( A e. On /\ L < L <
2 ssltex1
 |-  ( L < L e. _V )
3 1 2 syl
 |-  ( ( ( A e. On /\ L < L e. _V )
4 simprl
 |-  ( ( ( A e. On /\ L < L C_ ( _Old ` A ) )
5 3 4 elpwd
 |-  ( ( ( A e. On /\ L < L e. ~P ( _Old ` A ) )
6 ssltex2
 |-  ( L < R e. _V )
7 1 6 syl
 |-  ( ( ( A e. On /\ L < R e. _V )
8 simprr
 |-  ( ( ( A e. On /\ L < R C_ ( _Old ` A ) )
9 7 8 elpwd
 |-  ( ( ( A e. On /\ L < R e. ~P ( _Old ` A ) )
10 eqidd
 |-  ( ( ( A e. On /\ L < ( L |s R ) = ( L |s R ) )
11 breq1
 |-  ( l = L -> ( l < L <
12 oveq1
 |-  ( l = L -> ( l |s r ) = ( L |s r ) )
13 12 eqeq1d
 |-  ( l = L -> ( ( l |s r ) = ( L |s R ) <-> ( L |s r ) = ( L |s R ) ) )
14 11 13 anbi12d
 |-  ( l = L -> ( ( l < ( L <
15 breq2
 |-  ( r = R -> ( L < L <
16 oveq2
 |-  ( r = R -> ( L |s r ) = ( L |s R ) )
17 16 eqeq1d
 |-  ( r = R -> ( ( L |s r ) = ( L |s R ) <-> ( L |s R ) = ( L |s R ) ) )
18 15 17 anbi12d
 |-  ( r = R -> ( ( L < ( L <
19 14 18 rspc2ev
 |-  ( ( L e. ~P ( _Old ` A ) /\ R e. ~P ( _Old ` A ) /\ ( L < E. l e. ~P ( _Old ` A ) E. r e. ~P ( _Old ` A ) ( l <
20 5 9 1 10 19 syl112anc
 |-  ( ( ( A e. On /\ L < E. l e. ~P ( _Old ` A ) E. r e. ~P ( _Old ` A ) ( l <
21 elmade2
 |-  ( A e. On -> ( ( L |s R ) e. ( _M ` A ) <-> E. l e. ~P ( _Old ` A ) E. r e. ~P ( _Old ` A ) ( l <
22 21 ad2antrr
 |-  ( ( ( A e. On /\ L < ( ( L |s R ) e. ( _M ` A ) <-> E. l e. ~P ( _Old ` A ) E. r e. ~P ( _Old ` A ) ( l <
23 20 22 mpbird
 |-  ( ( ( A e. On /\ L < ( L |s R ) e. ( _M ` A ) )