Step |
Hyp |
Ref |
Expression |
1 |
|
pssss |
⊢ ( 𝐹 ⊊ 𝐺 → 𝐹 ⊆ 𝐺 ) |
2 |
|
dmss |
⊢ ( 𝐹 ⊆ 𝐺 → dom 𝐹 ⊆ dom 𝐺 ) |
3 |
1 2
|
syl |
⊢ ( 𝐹 ⊊ 𝐺 → dom 𝐹 ⊆ dom 𝐺 ) |
4 |
3
|
a1i |
⊢ ( Fun 𝐺 → ( 𝐹 ⊊ 𝐺 → dom 𝐹 ⊆ dom 𝐺 ) ) |
5 |
|
pssdif |
⊢ ( 𝐹 ⊊ 𝐺 → ( 𝐺 ∖ 𝐹 ) ≠ ∅ ) |
6 |
|
n0 |
⊢ ( ( 𝐺 ∖ 𝐹 ) ≠ ∅ ↔ ∃ 𝑝 𝑝 ∈ ( 𝐺 ∖ 𝐹 ) ) |
7 |
5 6
|
sylib |
⊢ ( 𝐹 ⊊ 𝐺 → ∃ 𝑝 𝑝 ∈ ( 𝐺 ∖ 𝐹 ) ) |
8 |
7
|
adantl |
⊢ ( ( Fun 𝐺 ∧ 𝐹 ⊊ 𝐺 ) → ∃ 𝑝 𝑝 ∈ ( 𝐺 ∖ 𝐹 ) ) |
9 |
|
funrel |
⊢ ( Fun 𝐺 → Rel 𝐺 ) |
10 |
|
reldif |
⊢ ( Rel 𝐺 → Rel ( 𝐺 ∖ 𝐹 ) ) |
11 |
9 10
|
syl |
⊢ ( Fun 𝐺 → Rel ( 𝐺 ∖ 𝐹 ) ) |
12 |
|
elrel |
⊢ ( ( Rel ( 𝐺 ∖ 𝐹 ) ∧ 𝑝 ∈ ( 𝐺 ∖ 𝐹 ) ) → ∃ 𝑥 ∃ 𝑦 𝑝 = 〈 𝑥 , 𝑦 〉 ) |
13 |
|
eleq1 |
⊢ ( 𝑝 = 〈 𝑥 , 𝑦 〉 → ( 𝑝 ∈ ( 𝐺 ∖ 𝐹 ) ↔ 〈 𝑥 , 𝑦 〉 ∈ ( 𝐺 ∖ 𝐹 ) ) ) |
14 |
|
df-br |
⊢ ( 𝑥 ( 𝐺 ∖ 𝐹 ) 𝑦 ↔ 〈 𝑥 , 𝑦 〉 ∈ ( 𝐺 ∖ 𝐹 ) ) |
15 |
13 14
|
bitr4di |
⊢ ( 𝑝 = 〈 𝑥 , 𝑦 〉 → ( 𝑝 ∈ ( 𝐺 ∖ 𝐹 ) ↔ 𝑥 ( 𝐺 ∖ 𝐹 ) 𝑦 ) ) |
16 |
15
|
biimpcd |
⊢ ( 𝑝 ∈ ( 𝐺 ∖ 𝐹 ) → ( 𝑝 = 〈 𝑥 , 𝑦 〉 → 𝑥 ( 𝐺 ∖ 𝐹 ) 𝑦 ) ) |
17 |
16
|
adantl |
⊢ ( ( Rel ( 𝐺 ∖ 𝐹 ) ∧ 𝑝 ∈ ( 𝐺 ∖ 𝐹 ) ) → ( 𝑝 = 〈 𝑥 , 𝑦 〉 → 𝑥 ( 𝐺 ∖ 𝐹 ) 𝑦 ) ) |
18 |
17
|
2eximdv |
⊢ ( ( Rel ( 𝐺 ∖ 𝐹 ) ∧ 𝑝 ∈ ( 𝐺 ∖ 𝐹 ) ) → ( ∃ 𝑥 ∃ 𝑦 𝑝 = 〈 𝑥 , 𝑦 〉 → ∃ 𝑥 ∃ 𝑦 𝑥 ( 𝐺 ∖ 𝐹 ) 𝑦 ) ) |
19 |
12 18
|
mpd |
⊢ ( ( Rel ( 𝐺 ∖ 𝐹 ) ∧ 𝑝 ∈ ( 𝐺 ∖ 𝐹 ) ) → ∃ 𝑥 ∃ 𝑦 𝑥 ( 𝐺 ∖ 𝐹 ) 𝑦 ) |
20 |
19
|
ex |
⊢ ( Rel ( 𝐺 ∖ 𝐹 ) → ( 𝑝 ∈ ( 𝐺 ∖ 𝐹 ) → ∃ 𝑥 ∃ 𝑦 𝑥 ( 𝐺 ∖ 𝐹 ) 𝑦 ) ) |
21 |
11 20
|
syl |
⊢ ( Fun 𝐺 → ( 𝑝 ∈ ( 𝐺 ∖ 𝐹 ) → ∃ 𝑥 ∃ 𝑦 𝑥 ( 𝐺 ∖ 𝐹 ) 𝑦 ) ) |
22 |
21
|
adantr |
⊢ ( ( Fun 𝐺 ∧ 𝐹 ⊊ 𝐺 ) → ( 𝑝 ∈ ( 𝐺 ∖ 𝐹 ) → ∃ 𝑥 ∃ 𝑦 𝑥 ( 𝐺 ∖ 𝐹 ) 𝑦 ) ) |
23 |
|
difss |
⊢ ( 𝐺 ∖ 𝐹 ) ⊆ 𝐺 |
24 |
23
|
ssbri |
⊢ ( 𝑥 ( 𝐺 ∖ 𝐹 ) 𝑦 → 𝑥 𝐺 𝑦 ) |
25 |
24
|
eximi |
⊢ ( ∃ 𝑦 𝑥 ( 𝐺 ∖ 𝐹 ) 𝑦 → ∃ 𝑦 𝑥 𝐺 𝑦 ) |
26 |
25
|
a1i |
⊢ ( ( Fun 𝐺 ∧ 𝐹 ⊊ 𝐺 ) → ( ∃ 𝑦 𝑥 ( 𝐺 ∖ 𝐹 ) 𝑦 → ∃ 𝑦 𝑥 𝐺 𝑦 ) ) |
27 |
|
brdif |
⊢ ( 𝑥 ( 𝐺 ∖ 𝐹 ) 𝑦 ↔ ( 𝑥 𝐺 𝑦 ∧ ¬ 𝑥 𝐹 𝑦 ) ) |
28 |
27
|
simprbi |
⊢ ( 𝑥 ( 𝐺 ∖ 𝐹 ) 𝑦 → ¬ 𝑥 𝐹 𝑦 ) |
29 |
28
|
adantl |
⊢ ( ( ( Fun 𝐺 ∧ 𝐹 ⊊ 𝐺 ) ∧ 𝑥 ( 𝐺 ∖ 𝐹 ) 𝑦 ) → ¬ 𝑥 𝐹 𝑦 ) |
30 |
1
|
ssbrd |
⊢ ( 𝐹 ⊊ 𝐺 → ( 𝑥 𝐹 𝑧 → 𝑥 𝐺 𝑧 ) ) |
31 |
30
|
ad2antlr |
⊢ ( ( ( Fun 𝐺 ∧ 𝐹 ⊊ 𝐺 ) ∧ 𝑥 ( 𝐺 ∖ 𝐹 ) 𝑦 ) → ( 𝑥 𝐹 𝑧 → 𝑥 𝐺 𝑧 ) ) |
32 |
|
dffun2 |
⊢ ( Fun 𝐺 ↔ ( Rel 𝐺 ∧ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝐺 𝑦 ∧ 𝑥 𝐺 𝑧 ) → 𝑦 = 𝑧 ) ) ) |
33 |
32
|
simprbi |
⊢ ( Fun 𝐺 → ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝐺 𝑦 ∧ 𝑥 𝐺 𝑧 ) → 𝑦 = 𝑧 ) ) |
34 |
|
2sp |
⊢ ( ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝐺 𝑦 ∧ 𝑥 𝐺 𝑧 ) → 𝑦 = 𝑧 ) → ( ( 𝑥 𝐺 𝑦 ∧ 𝑥 𝐺 𝑧 ) → 𝑦 = 𝑧 ) ) |
35 |
34
|
sps |
⊢ ( ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝐺 𝑦 ∧ 𝑥 𝐺 𝑧 ) → 𝑦 = 𝑧 ) → ( ( 𝑥 𝐺 𝑦 ∧ 𝑥 𝐺 𝑧 ) → 𝑦 = 𝑧 ) ) |
36 |
33 35
|
syl |
⊢ ( Fun 𝐺 → ( ( 𝑥 𝐺 𝑦 ∧ 𝑥 𝐺 𝑧 ) → 𝑦 = 𝑧 ) ) |
37 |
|
breq2 |
⊢ ( 𝑦 = 𝑧 → ( 𝑥 𝐹 𝑦 ↔ 𝑥 𝐹 𝑧 ) ) |
38 |
37
|
biimprd |
⊢ ( 𝑦 = 𝑧 → ( 𝑥 𝐹 𝑧 → 𝑥 𝐹 𝑦 ) ) |
39 |
36 38
|
syl6 |
⊢ ( Fun 𝐺 → ( ( 𝑥 𝐺 𝑦 ∧ 𝑥 𝐺 𝑧 ) → ( 𝑥 𝐹 𝑧 → 𝑥 𝐹 𝑦 ) ) ) |
40 |
39
|
expd |
⊢ ( Fun 𝐺 → ( 𝑥 𝐺 𝑦 → ( 𝑥 𝐺 𝑧 → ( 𝑥 𝐹 𝑧 → 𝑥 𝐹 𝑦 ) ) ) ) |
41 |
27
|
simplbi |
⊢ ( 𝑥 ( 𝐺 ∖ 𝐹 ) 𝑦 → 𝑥 𝐺 𝑦 ) |
42 |
40 41
|
impel |
⊢ ( ( Fun 𝐺 ∧ 𝑥 ( 𝐺 ∖ 𝐹 ) 𝑦 ) → ( 𝑥 𝐺 𝑧 → ( 𝑥 𝐹 𝑧 → 𝑥 𝐹 𝑦 ) ) ) |
43 |
42
|
adantlr |
⊢ ( ( ( Fun 𝐺 ∧ 𝐹 ⊊ 𝐺 ) ∧ 𝑥 ( 𝐺 ∖ 𝐹 ) 𝑦 ) → ( 𝑥 𝐺 𝑧 → ( 𝑥 𝐹 𝑧 → 𝑥 𝐹 𝑦 ) ) ) |
44 |
43
|
com23 |
⊢ ( ( ( Fun 𝐺 ∧ 𝐹 ⊊ 𝐺 ) ∧ 𝑥 ( 𝐺 ∖ 𝐹 ) 𝑦 ) → ( 𝑥 𝐹 𝑧 → ( 𝑥 𝐺 𝑧 → 𝑥 𝐹 𝑦 ) ) ) |
45 |
31 44
|
mpdd |
⊢ ( ( ( Fun 𝐺 ∧ 𝐹 ⊊ 𝐺 ) ∧ 𝑥 ( 𝐺 ∖ 𝐹 ) 𝑦 ) → ( 𝑥 𝐹 𝑧 → 𝑥 𝐹 𝑦 ) ) |
46 |
45
|
exlimdv |
⊢ ( ( ( Fun 𝐺 ∧ 𝐹 ⊊ 𝐺 ) ∧ 𝑥 ( 𝐺 ∖ 𝐹 ) 𝑦 ) → ( ∃ 𝑧 𝑥 𝐹 𝑧 → 𝑥 𝐹 𝑦 ) ) |
47 |
29 46
|
mtod |
⊢ ( ( ( Fun 𝐺 ∧ 𝐹 ⊊ 𝐺 ) ∧ 𝑥 ( 𝐺 ∖ 𝐹 ) 𝑦 ) → ¬ ∃ 𝑧 𝑥 𝐹 𝑧 ) |
48 |
47
|
ex |
⊢ ( ( Fun 𝐺 ∧ 𝐹 ⊊ 𝐺 ) → ( 𝑥 ( 𝐺 ∖ 𝐹 ) 𝑦 → ¬ ∃ 𝑧 𝑥 𝐹 𝑧 ) ) |
49 |
48
|
exlimdv |
⊢ ( ( Fun 𝐺 ∧ 𝐹 ⊊ 𝐺 ) → ( ∃ 𝑦 𝑥 ( 𝐺 ∖ 𝐹 ) 𝑦 → ¬ ∃ 𝑧 𝑥 𝐹 𝑧 ) ) |
50 |
26 49
|
jcad |
⊢ ( ( Fun 𝐺 ∧ 𝐹 ⊊ 𝐺 ) → ( ∃ 𝑦 𝑥 ( 𝐺 ∖ 𝐹 ) 𝑦 → ( ∃ 𝑦 𝑥 𝐺 𝑦 ∧ ¬ ∃ 𝑧 𝑥 𝐹 𝑧 ) ) ) |
51 |
50
|
eximdv |
⊢ ( ( Fun 𝐺 ∧ 𝐹 ⊊ 𝐺 ) → ( ∃ 𝑥 ∃ 𝑦 𝑥 ( 𝐺 ∖ 𝐹 ) 𝑦 → ∃ 𝑥 ( ∃ 𝑦 𝑥 𝐺 𝑦 ∧ ¬ ∃ 𝑧 𝑥 𝐹 𝑧 ) ) ) |
52 |
22 51
|
syld |
⊢ ( ( Fun 𝐺 ∧ 𝐹 ⊊ 𝐺 ) → ( 𝑝 ∈ ( 𝐺 ∖ 𝐹 ) → ∃ 𝑥 ( ∃ 𝑦 𝑥 𝐺 𝑦 ∧ ¬ ∃ 𝑧 𝑥 𝐹 𝑧 ) ) ) |
53 |
52
|
exlimdv |
⊢ ( ( Fun 𝐺 ∧ 𝐹 ⊊ 𝐺 ) → ( ∃ 𝑝 𝑝 ∈ ( 𝐺 ∖ 𝐹 ) → ∃ 𝑥 ( ∃ 𝑦 𝑥 𝐺 𝑦 ∧ ¬ ∃ 𝑧 𝑥 𝐹 𝑧 ) ) ) |
54 |
8 53
|
mpd |
⊢ ( ( Fun 𝐺 ∧ 𝐹 ⊊ 𝐺 ) → ∃ 𝑥 ( ∃ 𝑦 𝑥 𝐺 𝑦 ∧ ¬ ∃ 𝑧 𝑥 𝐹 𝑧 ) ) |
55 |
|
nss |
⊢ ( ¬ dom 𝐺 ⊆ dom 𝐹 ↔ ∃ 𝑥 ( 𝑥 ∈ dom 𝐺 ∧ ¬ 𝑥 ∈ dom 𝐹 ) ) |
56 |
|
vex |
⊢ 𝑥 ∈ V |
57 |
56
|
eldm |
⊢ ( 𝑥 ∈ dom 𝐺 ↔ ∃ 𝑦 𝑥 𝐺 𝑦 ) |
58 |
56
|
eldm |
⊢ ( 𝑥 ∈ dom 𝐹 ↔ ∃ 𝑧 𝑥 𝐹 𝑧 ) |
59 |
58
|
notbii |
⊢ ( ¬ 𝑥 ∈ dom 𝐹 ↔ ¬ ∃ 𝑧 𝑥 𝐹 𝑧 ) |
60 |
57 59
|
anbi12i |
⊢ ( ( 𝑥 ∈ dom 𝐺 ∧ ¬ 𝑥 ∈ dom 𝐹 ) ↔ ( ∃ 𝑦 𝑥 𝐺 𝑦 ∧ ¬ ∃ 𝑧 𝑥 𝐹 𝑧 ) ) |
61 |
60
|
exbii |
⊢ ( ∃ 𝑥 ( 𝑥 ∈ dom 𝐺 ∧ ¬ 𝑥 ∈ dom 𝐹 ) ↔ ∃ 𝑥 ( ∃ 𝑦 𝑥 𝐺 𝑦 ∧ ¬ ∃ 𝑧 𝑥 𝐹 𝑧 ) ) |
62 |
55 61
|
bitri |
⊢ ( ¬ dom 𝐺 ⊆ dom 𝐹 ↔ ∃ 𝑥 ( ∃ 𝑦 𝑥 𝐺 𝑦 ∧ ¬ ∃ 𝑧 𝑥 𝐹 𝑧 ) ) |
63 |
54 62
|
sylibr |
⊢ ( ( Fun 𝐺 ∧ 𝐹 ⊊ 𝐺 ) → ¬ dom 𝐺 ⊆ dom 𝐹 ) |
64 |
63
|
ex |
⊢ ( Fun 𝐺 → ( 𝐹 ⊊ 𝐺 → ¬ dom 𝐺 ⊆ dom 𝐹 ) ) |
65 |
4 64
|
jcad |
⊢ ( Fun 𝐺 → ( 𝐹 ⊊ 𝐺 → ( dom 𝐹 ⊆ dom 𝐺 ∧ ¬ dom 𝐺 ⊆ dom 𝐹 ) ) ) |
66 |
|
dfpss3 |
⊢ ( dom 𝐹 ⊊ dom 𝐺 ↔ ( dom 𝐹 ⊆ dom 𝐺 ∧ ¬ dom 𝐺 ⊆ dom 𝐹 ) ) |
67 |
65 66
|
syl6ibr |
⊢ ( Fun 𝐺 → ( 𝐹 ⊊ 𝐺 → dom 𝐹 ⊊ dom 𝐺 ) ) |