Step |
Hyp |
Ref |
Expression |
1 |
|
infxrlbrnmpt2.x |
|- F/ x ph |
2 |
|
infxrlbrnmpt2.b |
|- ( ( ph /\ x e. A ) -> B e. RR* ) |
3 |
|
infxrlbrnmpt2.c |
|- ( ph -> C e. A ) |
4 |
|
infxrlbrnmpt2.d |
|- ( ph -> D e. RR* ) |
5 |
|
infxrlbrnmpt2.e |
|- ( x = C -> B = D ) |
6 |
|
eqid |
|- ( x e. A |-> B ) = ( x e. A |-> B ) |
7 |
1 6 2
|
rnmptssd |
|- ( ph -> ran ( x e. A |-> B ) C_ RR* ) |
8 |
6 5
|
elrnmpt1s |
|- ( ( C e. A /\ D e. RR* ) -> D e. ran ( x e. A |-> B ) ) |
9 |
3 4 8
|
syl2anc |
|- ( ph -> D e. ran ( x e. A |-> B ) ) |
10 |
|
infxrlb |
|- ( ( ran ( x e. A |-> B ) C_ RR* /\ D e. ran ( x e. A |-> B ) ) -> inf ( ran ( x e. A |-> B ) , RR* , < ) <_ D ) |
11 |
7 9 10
|
syl2anc |
|- ( ph -> inf ( ran ( x e. A |-> B ) , RR* , < ) <_ D ) |