Step |
Hyp |
Ref |
Expression |
1 |
|
ax-hilex |
|- ~H e. _V |
2 |
1
|
elpw2 |
|- ( H e. ~P ~H <-> H C_ ~H ) |
3 |
|
3anass |
|- ( ( 0h e. H /\ ( +h " ( H X. H ) ) C_ H /\ ( .h " ( CC X. H ) ) C_ H ) <-> ( 0h e. H /\ ( ( +h " ( H X. H ) ) C_ H /\ ( .h " ( CC X. H ) ) C_ H ) ) ) |
4 |
2 3
|
anbi12i |
|- ( ( H e. ~P ~H /\ ( 0h e. H /\ ( +h " ( H X. H ) ) C_ H /\ ( .h " ( CC X. H ) ) C_ H ) ) <-> ( H C_ ~H /\ ( 0h e. H /\ ( ( +h " ( H X. H ) ) C_ H /\ ( .h " ( CC X. H ) ) C_ H ) ) ) ) |
5 |
|
eleq2 |
|- ( h = H -> ( 0h e. h <-> 0h e. H ) ) |
6 |
|
id |
|- ( h = H -> h = H ) |
7 |
6
|
sqxpeqd |
|- ( h = H -> ( h X. h ) = ( H X. H ) ) |
8 |
7
|
imaeq2d |
|- ( h = H -> ( +h " ( h X. h ) ) = ( +h " ( H X. H ) ) ) |
9 |
8 6
|
sseq12d |
|- ( h = H -> ( ( +h " ( h X. h ) ) C_ h <-> ( +h " ( H X. H ) ) C_ H ) ) |
10 |
|
xpeq2 |
|- ( h = H -> ( CC X. h ) = ( CC X. H ) ) |
11 |
10
|
imaeq2d |
|- ( h = H -> ( .h " ( CC X. h ) ) = ( .h " ( CC X. H ) ) ) |
12 |
11 6
|
sseq12d |
|- ( h = H -> ( ( .h " ( CC X. h ) ) C_ h <-> ( .h " ( CC X. H ) ) C_ H ) ) |
13 |
5 9 12
|
3anbi123d |
|- ( h = H -> ( ( 0h e. h /\ ( +h " ( h X. h ) ) C_ h /\ ( .h " ( CC X. h ) ) C_ h ) <-> ( 0h e. H /\ ( +h " ( H X. H ) ) C_ H /\ ( .h " ( CC X. H ) ) C_ H ) ) ) |
14 |
|
df-sh |
|- SH = { h e. ~P ~H | ( 0h e. h /\ ( +h " ( h X. h ) ) C_ h /\ ( .h " ( CC X. h ) ) C_ h ) } |
15 |
13 14
|
elrab2 |
|- ( H e. SH <-> ( H e. ~P ~H /\ ( 0h e. H /\ ( +h " ( H X. H ) ) C_ H /\ ( .h " ( CC X. H ) ) C_ H ) ) ) |
16 |
|
anass |
|- ( ( ( H C_ ~H /\ 0h e. H ) /\ ( ( +h " ( H X. H ) ) C_ H /\ ( .h " ( CC X. H ) ) C_ H ) ) <-> ( H C_ ~H /\ ( 0h e. H /\ ( ( +h " ( H X. H ) ) C_ H /\ ( .h " ( CC X. H ) ) C_ H ) ) ) ) |
17 |
4 15 16
|
3bitr4i |
|- ( H e. SH <-> ( ( H C_ ~H /\ 0h e. H ) /\ ( ( +h " ( H X. H ) ) C_ H /\ ( .h " ( CC X. H ) ) C_ H ) ) ) |