Step |
Hyp |
Ref |
Expression |
1 |
|
suprubrnmpt.x |
|- F/ x ph |
2 |
|
suprubrnmpt.b |
|- ( ( ph /\ x e. A ) -> B e. RR ) |
3 |
|
suprubrnmpt.e |
|- ( ph -> E. y e. RR A. x e. A B <_ y ) |
4 |
|
eqid |
|- ( x e. A |-> B ) = ( x e. A |-> B ) |
5 |
1 4 2
|
rnmptssd |
|- ( ph -> ran ( x e. A |-> B ) C_ RR ) |
6 |
5
|
adantr |
|- ( ( ph /\ x e. A ) -> ran ( x e. A |-> B ) C_ RR ) |
7 |
|
simpr |
|- ( ( ph /\ x e. A ) -> x e. A ) |
8 |
4
|
elrnmpt1 |
|- ( ( x e. A /\ B e. RR ) -> B e. ran ( x e. A |-> B ) ) |
9 |
7 2 8
|
syl2anc |
|- ( ( ph /\ x e. A ) -> B e. ran ( x e. A |-> B ) ) |
10 |
9
|
ne0d |
|- ( ( ph /\ x e. A ) -> ran ( x e. A |-> B ) =/= (/) ) |
11 |
1 3
|
rnmptbdd |
|- ( ph -> E. y e. RR A. w e. ran ( x e. A |-> B ) w <_ y ) |
12 |
11
|
adantr |
|- ( ( ph /\ x e. A ) -> E. y e. RR A. w e. ran ( x e. A |-> B ) w <_ y ) |
13 |
6 10 12 9
|
suprubd |
|- ( ( ph /\ x e. A ) -> B <_ sup ( ran ( x e. A |-> B ) , RR , < ) ) |