Step |
Hyp |
Ref |
Expression |
1 |
|
isssc.1 |
|- ( ph -> H Fn ( S X. S ) ) |
2 |
|
dmxpid |
|- dom ( S X. S ) = S |
3 |
1
|
fndmd |
|- ( ph -> dom H = ( S X. S ) ) |
4 |
3
|
adantr |
|- ( ( ph /\ H e. _V ) -> dom H = ( S X. S ) ) |
5 |
|
dmexg |
|- ( H e. _V -> dom H e. _V ) |
6 |
5
|
adantl |
|- ( ( ph /\ H e. _V ) -> dom H e. _V ) |
7 |
4 6
|
eqeltrrd |
|- ( ( ph /\ H e. _V ) -> ( S X. S ) e. _V ) |
8 |
7
|
dmexd |
|- ( ( ph /\ H e. _V ) -> dom ( S X. S ) e. _V ) |
9 |
2 8
|
eqeltrrid |
|- ( ( ph /\ H e. _V ) -> S e. _V ) |
10 |
|
sqxpexg |
|- ( S e. _V -> ( S X. S ) e. _V ) |
11 |
|
fnex |
|- ( ( H Fn ( S X. S ) /\ ( S X. S ) e. _V ) -> H e. _V ) |
12 |
1 10 11
|
syl2an |
|- ( ( ph /\ S e. _V ) -> H e. _V ) |
13 |
9 12
|
impbida |
|- ( ph -> ( H e. _V <-> S e. _V ) ) |