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Extended real and complex numbers, real and complex projective lines
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bj-imdirval2lem
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bj-imdirval2
Metamath Proof Explorer
Ascii
Unicode
Theorem
bj-imdirval2lem
Description:
Lemma for
bj-imdirval2
and
bj-iminvval2
.
(Contributed by
BJ
, 23-May-2024)
Ref
Expression
Hypotheses
bj-imdirval2lem.exa
⊢
φ
→
A
∈
U
bj-imdirval2lem.exb
⊢
φ
→
B
∈
V
Assertion
bj-imdirval2lem
⊢
φ
→
x
y
|
x
⊆
A
∧
y
⊆
B
∧
ψ
∈
V
Proof
Step
Hyp
Ref
Expression
1
bj-imdirval2lem.exa
⊢
φ
→
A
∈
U
2
bj-imdirval2lem.exb
⊢
φ
→
B
∈
V
3
1
pwexd
⊢
φ
→
𝒫
A
∈
V
4
2
pwexd
⊢
φ
→
𝒫
B
∈
V
5
simprl
⊢
φ
∧
x
⊆
A
∧
y
⊆
B
→
x
⊆
A
6
velpw
⊢
x
∈
𝒫
A
↔
x
⊆
A
7
5
6
sylibr
⊢
φ
∧
x
⊆
A
∧
y
⊆
B
→
x
∈
𝒫
A
8
simprr
⊢
φ
∧
x
⊆
A
∧
y
⊆
B
→
y
⊆
B
9
velpw
⊢
y
∈
𝒫
B
↔
y
⊆
B
10
8
9
sylibr
⊢
φ
∧
x
⊆
A
∧
y
⊆
B
→
y
∈
𝒫
B
11
3
4
7
10
opabex2
⊢
φ
→
x
y
|
x
⊆
A
∧
y
⊆
B
∈
V
12
simpl
⊢
x
⊆
A
∧
y
⊆
B
∧
ψ
→
x
⊆
A
∧
y
⊆
B
13
12
ssopab2i
⊢
x
y
|
x
⊆
A
∧
y
⊆
B
∧
ψ
⊆
x
y
|
x
⊆
A
∧
y
⊆
B
14
13
a1i
⊢
φ
→
x
y
|
x
⊆
A
∧
y
⊆
B
∧
ψ
⊆
x
y
|
x
⊆
A
∧
y
⊆
B
15
11
14
ssexd
⊢
φ
→
x
y
|
x
⊆
A
∧
y
⊆
B
∧
ψ
∈
V