Step |
Hyp |
Ref |
Expression |
1 |
|
satefv |
|- ( ( M e. V /\ U e. ( Fmla ` _om ) ) -> ( M SatE U ) = ( ( ( M Sat ( _E i^i ( M X. M ) ) ) ` _om ) ` U ) ) |
2 |
1
|
eleq2d |
|- ( ( M e. V /\ U e. ( Fmla ` _om ) ) -> ( S e. ( M SatE U ) <-> S e. ( ( ( M Sat ( _E i^i ( M X. M ) ) ) ` _om ) ` U ) ) ) |
3 |
|
simpl |
|- ( ( M e. V /\ U e. ( Fmla ` _om ) ) -> M e. V ) |
4 |
|
incom |
|- ( _E i^i ( M X. M ) ) = ( ( M X. M ) i^i _E ) |
5 |
|
sqxpexg |
|- ( M e. V -> ( M X. M ) e. _V ) |
6 |
5
|
adantr |
|- ( ( M e. V /\ U e. ( Fmla ` _om ) ) -> ( M X. M ) e. _V ) |
7 |
|
inex1g |
|- ( ( M X. M ) e. _V -> ( ( M X. M ) i^i _E ) e. _V ) |
8 |
6 7
|
syl |
|- ( ( M e. V /\ U e. ( Fmla ` _om ) ) -> ( ( M X. M ) i^i _E ) e. _V ) |
9 |
4 8
|
eqeltrid |
|- ( ( M e. V /\ U e. ( Fmla ` _om ) ) -> ( _E i^i ( M X. M ) ) e. _V ) |
10 |
3 9
|
jca |
|- ( ( M e. V /\ U e. ( Fmla ` _om ) ) -> ( M e. V /\ ( _E i^i ( M X. M ) ) e. _V ) ) |
11 |
10
|
adantr |
|- ( ( ( M e. V /\ U e. ( Fmla ` _om ) ) /\ S e. ( ( ( M Sat ( _E i^i ( M X. M ) ) ) ` _om ) ` U ) ) -> ( M e. V /\ ( _E i^i ( M X. M ) ) e. _V ) ) |
12 |
|
simpr |
|- ( ( M e. V /\ U e. ( Fmla ` _om ) ) -> U e. ( Fmla ` _om ) ) |
13 |
12
|
adantr |
|- ( ( ( M e. V /\ U e. ( Fmla ` _om ) ) /\ S e. ( ( ( M Sat ( _E i^i ( M X. M ) ) ) ` _om ) ` U ) ) -> U e. ( Fmla ` _om ) ) |
14 |
|
simpr |
|- ( ( ( M e. V /\ U e. ( Fmla ` _om ) ) /\ S e. ( ( ( M Sat ( _E i^i ( M X. M ) ) ) ` _om ) ` U ) ) -> S e. ( ( ( M Sat ( _E i^i ( M X. M ) ) ) ` _om ) ` U ) ) |
15 |
11 13 14
|
3jca |
|- ( ( ( M e. V /\ U e. ( Fmla ` _om ) ) /\ S e. ( ( ( M Sat ( _E i^i ( M X. M ) ) ) ` _om ) ` U ) ) -> ( ( M e. V /\ ( _E i^i ( M X. M ) ) e. _V ) /\ U e. ( Fmla ` _om ) /\ S e. ( ( ( M Sat ( _E i^i ( M X. M ) ) ) ` _om ) ` U ) ) ) |
16 |
15
|
ex |
|- ( ( M e. V /\ U e. ( Fmla ` _om ) ) -> ( S e. ( ( ( M Sat ( _E i^i ( M X. M ) ) ) ` _om ) ` U ) -> ( ( M e. V /\ ( _E i^i ( M X. M ) ) e. _V ) /\ U e. ( Fmla ` _om ) /\ S e. ( ( ( M Sat ( _E i^i ( M X. M ) ) ) ` _om ) ` U ) ) ) ) |
17 |
2 16
|
sylbid |
|- ( ( M e. V /\ U e. ( Fmla ` _om ) ) -> ( S e. ( M SatE U ) -> ( ( M e. V /\ ( _E i^i ( M X. M ) ) e. _V ) /\ U e. ( Fmla ` _om ) /\ S e. ( ( ( M Sat ( _E i^i ( M X. M ) ) ) ` _om ) ` U ) ) ) ) |
18 |
17
|
3impia |
|- ( ( M e. V /\ U e. ( Fmla ` _om ) /\ S e. ( M SatE U ) ) -> ( ( M e. V /\ ( _E i^i ( M X. M ) ) e. _V ) /\ U e. ( Fmla ` _om ) /\ S e. ( ( ( M Sat ( _E i^i ( M X. M ) ) ) ` _om ) ` U ) ) ) |
19 |
|
satfvel |
|- ( ( ( M e. V /\ ( _E i^i ( M X. M ) ) e. _V ) /\ U e. ( Fmla ` _om ) /\ S e. ( ( ( M Sat ( _E i^i ( M X. M ) ) ) ` _om ) ` U ) ) -> S : _om --> M ) |
20 |
18 19
|
syl |
|- ( ( M e. V /\ U e. ( Fmla ` _om ) /\ S e. ( M SatE U ) ) -> S : _om --> M ) |