Step |
Hyp |
Ref |
Expression |
1 |
|
mapss2.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
2 |
|
mapss2.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑊 ) |
3 |
|
mapss2.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝑍 ) |
4 |
|
mapss2.n |
⊢ ( 𝜑 → 𝐶 ≠ ∅ ) |
5 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ⊆ 𝐵 ) → 𝐵 ∈ 𝑊 ) |
6 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐴 ⊆ 𝐵 ) → 𝐴 ⊆ 𝐵 ) |
7 |
|
mapss |
⊢ ( ( 𝐵 ∈ 𝑊 ∧ 𝐴 ⊆ 𝐵 ) → ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) ) |
8 |
5 6 7
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝐴 ⊆ 𝐵 ) → ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) ) |
9 |
8
|
ex |
⊢ ( 𝜑 → ( 𝐴 ⊆ 𝐵 → ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) ) ) |
10 |
|
n0 |
⊢ ( 𝐶 ≠ ∅ ↔ ∃ 𝑥 𝑥 ∈ 𝐶 ) |
11 |
4 10
|
sylib |
⊢ ( 𝜑 → ∃ 𝑥 𝑥 ∈ 𝐶 ) |
12 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) ) → ∃ 𝑥 𝑥 ∈ 𝐶 ) |
13 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ) → ( 𝑤 ∈ 𝐶 ↦ 𝑦 ) = ( 𝑤 ∈ 𝐶 ↦ 𝑦 ) ) |
14 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ) ∧ 𝑤 = 𝑥 ) → 𝑦 = 𝑦 ) |
15 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ) → 𝑥 ∈ 𝐶 ) |
16 |
|
vex |
⊢ 𝑦 ∈ V |
17 |
16
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ) → 𝑦 ∈ V ) |
18 |
13 14 15 17
|
fvmptd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ) → ( ( 𝑤 ∈ 𝐶 ↦ 𝑦 ) ‘ 𝑥 ) = 𝑦 ) |
19 |
18
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ) → 𝑦 = ( ( 𝑤 ∈ 𝐶 ↦ 𝑦 ) ‘ 𝑥 ) ) |
20 |
19
|
ad4ant13 |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) ) ∧ 𝑥 ∈ 𝐶 ) ∧ 𝑦 ∈ 𝐴 ) → 𝑦 = ( ( 𝑤 ∈ 𝐶 ↦ 𝑦 ) ‘ 𝑥 ) ) |
21 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) ) |
22 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) ∧ 𝑤 ∈ 𝐶 ) → 𝑦 ∈ 𝐴 ) |
23 |
22
|
fmpttd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → ( 𝑤 ∈ 𝐶 ↦ 𝑦 ) : 𝐶 ⟶ 𝐴 ) |
24 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → 𝐴 ∈ 𝑉 ) |
25 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → 𝐶 ∈ 𝑍 ) |
26 |
24 25
|
elmapd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → ( ( 𝑤 ∈ 𝐶 ↦ 𝑦 ) ∈ ( 𝐴 ↑m 𝐶 ) ↔ ( 𝑤 ∈ 𝐶 ↦ 𝑦 ) : 𝐶 ⟶ 𝐴 ) ) |
27 |
23 26
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → ( 𝑤 ∈ 𝐶 ↦ 𝑦 ) ∈ ( 𝐴 ↑m 𝐶 ) ) |
28 |
27
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝑤 ∈ 𝐶 ↦ 𝑦 ) ∈ ( 𝐴 ↑m 𝐶 ) ) |
29 |
21 28
|
sseldd |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝑤 ∈ 𝐶 ↦ 𝑦 ) ∈ ( 𝐵 ↑m 𝐶 ) ) |
30 |
|
elmapi |
⊢ ( ( 𝑤 ∈ 𝐶 ↦ 𝑦 ) ∈ ( 𝐵 ↑m 𝐶 ) → ( 𝑤 ∈ 𝐶 ↦ 𝑦 ) : 𝐶 ⟶ 𝐵 ) |
31 |
29 30
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝑤 ∈ 𝐶 ↦ 𝑦 ) : 𝐶 ⟶ 𝐵 ) |
32 |
31
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) ) ∧ 𝑥 ∈ 𝐶 ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝑤 ∈ 𝐶 ↦ 𝑦 ) : 𝐶 ⟶ 𝐵 ) |
33 |
|
simplr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) ) ∧ 𝑥 ∈ 𝐶 ) ∧ 𝑦 ∈ 𝐴 ) → 𝑥 ∈ 𝐶 ) |
34 |
32 33
|
ffvelrnd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) ) ∧ 𝑥 ∈ 𝐶 ) ∧ 𝑦 ∈ 𝐴 ) → ( ( 𝑤 ∈ 𝐶 ↦ 𝑦 ) ‘ 𝑥 ) ∈ 𝐵 ) |
35 |
20 34
|
eqeltrd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) ) ∧ 𝑥 ∈ 𝐶 ) ∧ 𝑦 ∈ 𝐴 ) → 𝑦 ∈ 𝐵 ) |
36 |
35
|
ralrimiva |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) ) ∧ 𝑥 ∈ 𝐶 ) → ∀ 𝑦 ∈ 𝐴 𝑦 ∈ 𝐵 ) |
37 |
|
dfss3 |
⊢ ( 𝐴 ⊆ 𝐵 ↔ ∀ 𝑦 ∈ 𝐴 𝑦 ∈ 𝐵 ) |
38 |
36 37
|
sylibr |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) ) ∧ 𝑥 ∈ 𝐶 ) → 𝐴 ⊆ 𝐵 ) |
39 |
38
|
ex |
⊢ ( ( 𝜑 ∧ ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) ) → ( 𝑥 ∈ 𝐶 → 𝐴 ⊆ 𝐵 ) ) |
40 |
39
|
exlimdv |
⊢ ( ( 𝜑 ∧ ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) ) → ( ∃ 𝑥 𝑥 ∈ 𝐶 → 𝐴 ⊆ 𝐵 ) ) |
41 |
12 40
|
mpd |
⊢ ( ( 𝜑 ∧ ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) ) → 𝐴 ⊆ 𝐵 ) |
42 |
41
|
ex |
⊢ ( 𝜑 → ( ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) → 𝐴 ⊆ 𝐵 ) ) |
43 |
9 42
|
impbid |
⊢ ( 𝜑 → ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) ) ) |