Step |
Hyp |
Ref |
Expression |
1 |
|
difmap.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
2 |
|
difmap.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑊 ) |
3 |
|
difmap.v |
⊢ ( 𝜑 → 𝐶 ∈ 𝑍 ) |
4 |
|
difmap.n |
⊢ ( 𝜑 → 𝐶 ≠ ∅ ) |
5 |
|
difssd |
⊢ ( 𝜑 → ( 𝐴 ∖ 𝐵 ) ⊆ 𝐴 ) |
6 |
|
mapss |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ( 𝐴 ∖ 𝐵 ) ⊆ 𝐴 ) → ( ( 𝐴 ∖ 𝐵 ) ↑m 𝐶 ) ⊆ ( 𝐴 ↑m 𝐶 ) ) |
7 |
1 5 6
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝐴 ∖ 𝐵 ) ↑m 𝐶 ) ⊆ ( 𝐴 ↑m 𝐶 ) ) |
8 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( ( 𝐴 ∖ 𝐵 ) ↑m 𝐶 ) ) → ( ( 𝐴 ∖ 𝐵 ) ↑m 𝐶 ) ⊆ ( 𝐴 ↑m 𝐶 ) ) |
9 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( ( 𝐴 ∖ 𝐵 ) ↑m 𝐶 ) ) → 𝑓 ∈ ( ( 𝐴 ∖ 𝐵 ) ↑m 𝐶 ) ) |
10 |
8 9
|
sseldd |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( ( 𝐴 ∖ 𝐵 ) ↑m 𝐶 ) ) → 𝑓 ∈ ( 𝐴 ↑m 𝐶 ) ) |
11 |
|
n0 |
⊢ ( 𝐶 ≠ ∅ ↔ ∃ 𝑥 𝑥 ∈ 𝐶 ) |
12 |
4 11
|
sylib |
⊢ ( 𝜑 → ∃ 𝑥 𝑥 ∈ 𝐶 ) |
13 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( ( 𝐴 ∖ 𝐵 ) ↑m 𝐶 ) ) → ∃ 𝑥 𝑥 ∈ 𝐶 ) |
14 |
|
simpr |
⊢ ( ( 𝑥 ∈ 𝐶 ∧ 𝑓 : 𝐶 ⟶ 𝐵 ) → 𝑓 : 𝐶 ⟶ 𝐵 ) |
15 |
|
simpl |
⊢ ( ( 𝑥 ∈ 𝐶 ∧ 𝑓 : 𝐶 ⟶ 𝐵 ) → 𝑥 ∈ 𝐶 ) |
16 |
14 15
|
ffvelrnd |
⊢ ( ( 𝑥 ∈ 𝐶 ∧ 𝑓 : 𝐶 ⟶ 𝐵 ) → ( 𝑓 ‘ 𝑥 ) ∈ 𝐵 ) |
17 |
16
|
adantll |
⊢ ( ( ( ( 𝜑 ∧ 𝑓 ∈ ( ( 𝐴 ∖ 𝐵 ) ↑m 𝐶 ) ) ∧ 𝑥 ∈ 𝐶 ) ∧ 𝑓 : 𝐶 ⟶ 𝐵 ) → ( 𝑓 ‘ 𝑥 ) ∈ 𝐵 ) |
18 |
|
elmapi |
⊢ ( 𝑓 ∈ ( ( 𝐴 ∖ 𝐵 ) ↑m 𝐶 ) → 𝑓 : 𝐶 ⟶ ( 𝐴 ∖ 𝐵 ) ) |
19 |
18
|
adantr |
⊢ ( ( 𝑓 ∈ ( ( 𝐴 ∖ 𝐵 ) ↑m 𝐶 ) ∧ 𝑥 ∈ 𝐶 ) → 𝑓 : 𝐶 ⟶ ( 𝐴 ∖ 𝐵 ) ) |
20 |
|
simpr |
⊢ ( ( 𝑓 ∈ ( ( 𝐴 ∖ 𝐵 ) ↑m 𝐶 ) ∧ 𝑥 ∈ 𝐶 ) → 𝑥 ∈ 𝐶 ) |
21 |
19 20
|
ffvelrnd |
⊢ ( ( 𝑓 ∈ ( ( 𝐴 ∖ 𝐵 ) ↑m 𝐶 ) ∧ 𝑥 ∈ 𝐶 ) → ( 𝑓 ‘ 𝑥 ) ∈ ( 𝐴 ∖ 𝐵 ) ) |
22 |
|
eldifn |
⊢ ( ( 𝑓 ‘ 𝑥 ) ∈ ( 𝐴 ∖ 𝐵 ) → ¬ ( 𝑓 ‘ 𝑥 ) ∈ 𝐵 ) |
23 |
21 22
|
syl |
⊢ ( ( 𝑓 ∈ ( ( 𝐴 ∖ 𝐵 ) ↑m 𝐶 ) ∧ 𝑥 ∈ 𝐶 ) → ¬ ( 𝑓 ‘ 𝑥 ) ∈ 𝐵 ) |
24 |
23
|
ad4ant23 |
⊢ ( ( ( ( 𝜑 ∧ 𝑓 ∈ ( ( 𝐴 ∖ 𝐵 ) ↑m 𝐶 ) ) ∧ 𝑥 ∈ 𝐶 ) ∧ 𝑓 : 𝐶 ⟶ 𝐵 ) → ¬ ( 𝑓 ‘ 𝑥 ) ∈ 𝐵 ) |
25 |
17 24
|
pm2.65da |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ ( ( 𝐴 ∖ 𝐵 ) ↑m 𝐶 ) ) ∧ 𝑥 ∈ 𝐶 ) → ¬ 𝑓 : 𝐶 ⟶ 𝐵 ) |
26 |
25
|
ex |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( ( 𝐴 ∖ 𝐵 ) ↑m 𝐶 ) ) → ( 𝑥 ∈ 𝐶 → ¬ 𝑓 : 𝐶 ⟶ 𝐵 ) ) |
27 |
26
|
exlimdv |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( ( 𝐴 ∖ 𝐵 ) ↑m 𝐶 ) ) → ( ∃ 𝑥 𝑥 ∈ 𝐶 → ¬ 𝑓 : 𝐶 ⟶ 𝐵 ) ) |
28 |
13 27
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( ( 𝐴 ∖ 𝐵 ) ↑m 𝐶 ) ) → ¬ 𝑓 : 𝐶 ⟶ 𝐵 ) |
29 |
|
elmapg |
⊢ ( ( 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑍 ) → ( 𝑓 ∈ ( 𝐵 ↑m 𝐶 ) ↔ 𝑓 : 𝐶 ⟶ 𝐵 ) ) |
30 |
2 3 29
|
syl2anc |
⊢ ( 𝜑 → ( 𝑓 ∈ ( 𝐵 ↑m 𝐶 ) ↔ 𝑓 : 𝐶 ⟶ 𝐵 ) ) |
31 |
30
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( ( 𝐴 ∖ 𝐵 ) ↑m 𝐶 ) ) → ( 𝑓 ∈ ( 𝐵 ↑m 𝐶 ) ↔ 𝑓 : 𝐶 ⟶ 𝐵 ) ) |
32 |
28 31
|
mtbird |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( ( 𝐴 ∖ 𝐵 ) ↑m 𝐶 ) ) → ¬ 𝑓 ∈ ( 𝐵 ↑m 𝐶 ) ) |
33 |
10 32
|
eldifd |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( ( 𝐴 ∖ 𝐵 ) ↑m 𝐶 ) ) → 𝑓 ∈ ( ( 𝐴 ↑m 𝐶 ) ∖ ( 𝐵 ↑m 𝐶 ) ) ) |
34 |
33
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑓 ∈ ( ( 𝐴 ∖ 𝐵 ) ↑m 𝐶 ) 𝑓 ∈ ( ( 𝐴 ↑m 𝐶 ) ∖ ( 𝐵 ↑m 𝐶 ) ) ) |
35 |
|
dfss3 |
⊢ ( ( ( 𝐴 ∖ 𝐵 ) ↑m 𝐶 ) ⊆ ( ( 𝐴 ↑m 𝐶 ) ∖ ( 𝐵 ↑m 𝐶 ) ) ↔ ∀ 𝑓 ∈ ( ( 𝐴 ∖ 𝐵 ) ↑m 𝐶 ) 𝑓 ∈ ( ( 𝐴 ↑m 𝐶 ) ∖ ( 𝐵 ↑m 𝐶 ) ) ) |
36 |
34 35
|
sylibr |
⊢ ( 𝜑 → ( ( 𝐴 ∖ 𝐵 ) ↑m 𝐶 ) ⊆ ( ( 𝐴 ↑m 𝐶 ) ∖ ( 𝐵 ↑m 𝐶 ) ) ) |