Step |
Hyp |
Ref |
Expression |
1 |
|
smfpimgtmpt.x |
|- F/ x ph |
2 |
|
smfpimgtmpt.s |
|- ( ph -> S e. SAlg ) |
3 |
|
smfpimgtmpt.b |
|- ( ( ph /\ x e. A ) -> B e. V ) |
4 |
|
smfpimgtmpt.f |
|- ( ph -> ( x e. A |-> B ) e. ( SMblFn ` S ) ) |
5 |
|
smfpimgtmpt.l |
|- ( ph -> L e. RR ) |
6 |
|
nfmpt1 |
|- F/_ x ( x e. A |-> B ) |
7 |
|
eqid |
|- dom ( x e. A |-> B ) = dom ( x e. A |-> B ) |
8 |
6 2 4 7 5
|
smfpreimagtf |
|- ( ph -> { x e. dom ( x e. A |-> B ) | L < ( ( x e. A |-> B ) ` x ) } e. ( S |`t dom ( x e. A |-> B ) ) ) |
9 |
|
eqid |
|- ( x e. A |-> B ) = ( x e. A |-> B ) |
10 |
1 9 3
|
dmmptdf |
|- ( ph -> dom ( x e. A |-> B ) = A ) |
11 |
6
|
nfdm |
|- F/_ x dom ( x e. A |-> B ) |
12 |
|
nfcv |
|- F/_ x A |
13 |
11 12
|
rabeqf |
|- ( dom ( x e. A |-> B ) = A -> { x e. dom ( x e. A |-> B ) | L < ( ( x e. A |-> B ) ` x ) } = { x e. A | L < ( ( x e. A |-> B ) ` x ) } ) |
14 |
10 13
|
syl |
|- ( ph -> { x e. dom ( x e. A |-> B ) | L < ( ( x e. A |-> B ) ` x ) } = { x e. A | L < ( ( x e. A |-> B ) ` x ) } ) |
15 |
9
|
a1i |
|- ( ph -> ( x e. A |-> B ) = ( x e. A |-> B ) ) |
16 |
15 3
|
fvmpt2d |
|- ( ( ph /\ x e. A ) -> ( ( x e. A |-> B ) ` x ) = B ) |
17 |
16
|
breq2d |
|- ( ( ph /\ x e. A ) -> ( L < ( ( x e. A |-> B ) ` x ) <-> L < B ) ) |
18 |
1 17
|
rabbida |
|- ( ph -> { x e. A | L < ( ( x e. A |-> B ) ` x ) } = { x e. A | L < B } ) |
19 |
|
eqidd |
|- ( ph -> { x e. A | L < B } = { x e. A | L < B } ) |
20 |
14 18 19
|
3eqtrrd |
|- ( ph -> { x e. A | L < B } = { x e. dom ( x e. A |-> B ) | L < ( ( x e. A |-> B ) ` x ) } ) |
21 |
10
|
eqcomd |
|- ( ph -> A = dom ( x e. A |-> B ) ) |
22 |
21
|
oveq2d |
|- ( ph -> ( S |`t A ) = ( S |`t dom ( x e. A |-> B ) ) ) |
23 |
20 22
|
eleq12d |
|- ( ph -> ( { x e. A | L < B } e. ( S |`t A ) <-> { x e. dom ( x e. A |-> B ) | L < ( ( x e. A |-> B ) ` x ) } e. ( S |`t dom ( x e. A |-> B ) ) ) ) |
24 |
8 23
|
mpbird |
|- ( ph -> { x e. A | L < B } e. ( S |`t A ) ) |