Step |
Hyp |
Ref |
Expression |
1 |
|
velsn |
|- ( x e. { A } <-> x = A ) |
2 |
|
eqimss |
|- ( x = A -> x C_ A ) |
3 |
2
|
pm4.71ri |
|- ( x = A <-> ( x C_ A /\ x = A ) ) |
4 |
1 3
|
bitri |
|- ( x e. { A } <-> ( x C_ A /\ x = A ) ) |
5 |
4
|
a1i |
|- ( ( A e. B /\ A =/= (/) ) -> ( x e. { A } <-> ( x C_ A /\ x = A ) ) ) |
6 |
|
simpl |
|- ( ( A e. B /\ A =/= (/) ) -> A e. B ) |
7 |
|
eqid |
|- A = A |
8 |
|
eqsbc3 |
|- ( A e. B -> ( [. A / x ]. x = A <-> A = A ) ) |
9 |
7 8
|
mpbiri |
|- ( A e. B -> [. A / x ]. x = A ) |
10 |
9
|
adantr |
|- ( ( A e. B /\ A =/= (/) ) -> [. A / x ]. x = A ) |
11 |
|
simpr |
|- ( ( A e. B /\ A =/= (/) ) -> A =/= (/) ) |
12 |
11
|
necomd |
|- ( ( A e. B /\ A =/= (/) ) -> (/) =/= A ) |
13 |
12
|
neneqd |
|- ( ( A e. B /\ A =/= (/) ) -> -. (/) = A ) |
14 |
|
0ex |
|- (/) e. _V |
15 |
|
eqsbc3 |
|- ( (/) e. _V -> ( [. (/) / x ]. x = A <-> (/) = A ) ) |
16 |
14 15
|
ax-mp |
|- ( [. (/) / x ]. x = A <-> (/) = A ) |
17 |
13 16
|
sylnibr |
|- ( ( A e. B /\ A =/= (/) ) -> -. [. (/) / x ]. x = A ) |
18 |
|
sseq1 |
|- ( x = A -> ( x C_ y <-> A C_ y ) ) |
19 |
18
|
anbi2d |
|- ( x = A -> ( ( y C_ A /\ x C_ y ) <-> ( y C_ A /\ A C_ y ) ) ) |
20 |
|
eqss |
|- ( y = A <-> ( y C_ A /\ A C_ y ) ) |
21 |
20
|
biimpri |
|- ( ( y C_ A /\ A C_ y ) -> y = A ) |
22 |
19 21
|
syl6bi |
|- ( x = A -> ( ( y C_ A /\ x C_ y ) -> y = A ) ) |
23 |
22
|
com12 |
|- ( ( y C_ A /\ x C_ y ) -> ( x = A -> y = A ) ) |
24 |
23
|
3adant1 |
|- ( ( ( A e. B /\ A =/= (/) ) /\ y C_ A /\ x C_ y ) -> ( x = A -> y = A ) ) |
25 |
|
sbcid |
|- ( [. x / x ]. x = A <-> x = A ) |
26 |
|
eqsbc3 |
|- ( y e. _V -> ( [. y / x ]. x = A <-> y = A ) ) |
27 |
26
|
elv |
|- ( [. y / x ]. x = A <-> y = A ) |
28 |
24 25 27
|
3imtr4g |
|- ( ( ( A e. B /\ A =/= (/) ) /\ y C_ A /\ x C_ y ) -> ( [. x / x ]. x = A -> [. y / x ]. x = A ) ) |
29 |
|
ineq12 |
|- ( ( y = A /\ x = A ) -> ( y i^i x ) = ( A i^i A ) ) |
30 |
|
inidm |
|- ( A i^i A ) = A |
31 |
29 30
|
eqtrdi |
|- ( ( y = A /\ x = A ) -> ( y i^i x ) = A ) |
32 |
27 25 31
|
syl2anb |
|- ( ( [. y / x ]. x = A /\ [. x / x ]. x = A ) -> ( y i^i x ) = A ) |
33 |
|
vex |
|- y e. _V |
34 |
33
|
inex1 |
|- ( y i^i x ) e. _V |
35 |
|
eqsbc3 |
|- ( ( y i^i x ) e. _V -> ( [. ( y i^i x ) / x ]. x = A <-> ( y i^i x ) = A ) ) |
36 |
34 35
|
ax-mp |
|- ( [. ( y i^i x ) / x ]. x = A <-> ( y i^i x ) = A ) |
37 |
32 36
|
sylibr |
|- ( ( [. y / x ]. x = A /\ [. x / x ]. x = A ) -> [. ( y i^i x ) / x ]. x = A ) |
38 |
37
|
a1i |
|- ( ( ( A e. B /\ A =/= (/) ) /\ y C_ A /\ x C_ A ) -> ( ( [. y / x ]. x = A /\ [. x / x ]. x = A ) -> [. ( y i^i x ) / x ]. x = A ) ) |
39 |
5 6 10 17 28 38
|
isfild |
|- ( ( A e. B /\ A =/= (/) ) -> { A } e. ( Fil ` A ) ) |