Step |
Hyp |
Ref |
Expression |
1 |
|
f1oweALT.1 |
⊢ 𝑅 = { 〈 𝑥 , 𝑦 〉 ∣ ( 𝐹 ‘ 𝑥 ) 𝑆 ( 𝐹 ‘ 𝑦 ) } |
2 |
|
f1ofo |
⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 → 𝐹 : 𝐴 –onto→ 𝐵 ) |
3 |
|
df-fo |
⊢ ( 𝐹 : 𝐴 –onto→ 𝐵 ↔ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ) ) |
4 |
|
freq2 |
⊢ ( ran 𝐹 = 𝐵 → ( 𝑆 Fr ran 𝐹 ↔ 𝑆 Fr 𝐵 ) ) |
5 |
4
|
biimprd |
⊢ ( ran 𝐹 = 𝐵 → ( 𝑆 Fr 𝐵 → 𝑆 Fr ran 𝐹 ) ) |
6 |
|
df-fn |
⊢ ( 𝐹 Fn 𝐴 ↔ ( Fun 𝐹 ∧ dom 𝐹 = 𝐴 ) ) |
7 |
|
df-fr |
⊢ ( 𝑆 Fr ran 𝐹 ↔ ∀ 𝑤 ( ( 𝑤 ⊆ ran 𝐹 ∧ 𝑤 ≠ ∅ ) → ∃ 𝑢 ∈ 𝑤 ∀ 𝑓 ∈ 𝑤 ¬ 𝑓 𝑆 𝑢 ) ) |
8 |
|
vex |
⊢ 𝑧 ∈ V |
9 |
8
|
funimaex |
⊢ ( Fun 𝐹 → ( 𝐹 “ 𝑧 ) ∈ V ) |
10 |
|
n0 |
⊢ ( 𝑧 ≠ ∅ ↔ ∃ 𝑤 𝑤 ∈ 𝑧 ) |
11 |
|
funfvima2 |
⊢ ( ( Fun 𝐹 ∧ 𝑧 ⊆ dom 𝐹 ) → ( 𝑤 ∈ 𝑧 → ( 𝐹 ‘ 𝑤 ) ∈ ( 𝐹 “ 𝑧 ) ) ) |
12 |
|
ne0i |
⊢ ( ( 𝐹 ‘ 𝑤 ) ∈ ( 𝐹 “ 𝑧 ) → ( 𝐹 “ 𝑧 ) ≠ ∅ ) |
13 |
11 12
|
syl6 |
⊢ ( ( Fun 𝐹 ∧ 𝑧 ⊆ dom 𝐹 ) → ( 𝑤 ∈ 𝑧 → ( 𝐹 “ 𝑧 ) ≠ ∅ ) ) |
14 |
13
|
exlimdv |
⊢ ( ( Fun 𝐹 ∧ 𝑧 ⊆ dom 𝐹 ) → ( ∃ 𝑤 𝑤 ∈ 𝑧 → ( 𝐹 “ 𝑧 ) ≠ ∅ ) ) |
15 |
10 14
|
syl5bi |
⊢ ( ( Fun 𝐹 ∧ 𝑧 ⊆ dom 𝐹 ) → ( 𝑧 ≠ ∅ → ( 𝐹 “ 𝑧 ) ≠ ∅ ) ) |
16 |
15
|
imp |
⊢ ( ( ( Fun 𝐹 ∧ 𝑧 ⊆ dom 𝐹 ) ∧ 𝑧 ≠ ∅ ) → ( 𝐹 “ 𝑧 ) ≠ ∅ ) |
17 |
|
imassrn |
⊢ ( 𝐹 “ 𝑧 ) ⊆ ran 𝐹 |
18 |
16 17
|
jctil |
⊢ ( ( ( Fun 𝐹 ∧ 𝑧 ⊆ dom 𝐹 ) ∧ 𝑧 ≠ ∅ ) → ( ( 𝐹 “ 𝑧 ) ⊆ ran 𝐹 ∧ ( 𝐹 “ 𝑧 ) ≠ ∅ ) ) |
19 |
|
sseq1 |
⊢ ( 𝑤 = ( 𝐹 “ 𝑧 ) → ( 𝑤 ⊆ ran 𝐹 ↔ ( 𝐹 “ 𝑧 ) ⊆ ran 𝐹 ) ) |
20 |
|
neeq1 |
⊢ ( 𝑤 = ( 𝐹 “ 𝑧 ) → ( 𝑤 ≠ ∅ ↔ ( 𝐹 “ 𝑧 ) ≠ ∅ ) ) |
21 |
19 20
|
anbi12d |
⊢ ( 𝑤 = ( 𝐹 “ 𝑧 ) → ( ( 𝑤 ⊆ ran 𝐹 ∧ 𝑤 ≠ ∅ ) ↔ ( ( 𝐹 “ 𝑧 ) ⊆ ran 𝐹 ∧ ( 𝐹 “ 𝑧 ) ≠ ∅ ) ) ) |
22 |
|
raleq |
⊢ ( 𝑤 = ( 𝐹 “ 𝑧 ) → ( ∀ 𝑓 ∈ 𝑤 ¬ 𝑓 𝑆 𝑢 ↔ ∀ 𝑓 ∈ ( 𝐹 “ 𝑧 ) ¬ 𝑓 𝑆 𝑢 ) ) |
23 |
22
|
rexeqbi1dv |
⊢ ( 𝑤 = ( 𝐹 “ 𝑧 ) → ( ∃ 𝑢 ∈ 𝑤 ∀ 𝑓 ∈ 𝑤 ¬ 𝑓 𝑆 𝑢 ↔ ∃ 𝑢 ∈ ( 𝐹 “ 𝑧 ) ∀ 𝑓 ∈ ( 𝐹 “ 𝑧 ) ¬ 𝑓 𝑆 𝑢 ) ) |
24 |
21 23
|
imbi12d |
⊢ ( 𝑤 = ( 𝐹 “ 𝑧 ) → ( ( ( 𝑤 ⊆ ran 𝐹 ∧ 𝑤 ≠ ∅ ) → ∃ 𝑢 ∈ 𝑤 ∀ 𝑓 ∈ 𝑤 ¬ 𝑓 𝑆 𝑢 ) ↔ ( ( ( 𝐹 “ 𝑧 ) ⊆ ran 𝐹 ∧ ( 𝐹 “ 𝑧 ) ≠ ∅ ) → ∃ 𝑢 ∈ ( 𝐹 “ 𝑧 ) ∀ 𝑓 ∈ ( 𝐹 “ 𝑧 ) ¬ 𝑓 𝑆 𝑢 ) ) ) |
25 |
24
|
spcgv |
⊢ ( ( 𝐹 “ 𝑧 ) ∈ V → ( ∀ 𝑤 ( ( 𝑤 ⊆ ran 𝐹 ∧ 𝑤 ≠ ∅ ) → ∃ 𝑢 ∈ 𝑤 ∀ 𝑓 ∈ 𝑤 ¬ 𝑓 𝑆 𝑢 ) → ( ( ( 𝐹 “ 𝑧 ) ⊆ ran 𝐹 ∧ ( 𝐹 “ 𝑧 ) ≠ ∅ ) → ∃ 𝑢 ∈ ( 𝐹 “ 𝑧 ) ∀ 𝑓 ∈ ( 𝐹 “ 𝑧 ) ¬ 𝑓 𝑆 𝑢 ) ) ) |
26 |
18 25
|
syl7 |
⊢ ( ( 𝐹 “ 𝑧 ) ∈ V → ( ∀ 𝑤 ( ( 𝑤 ⊆ ran 𝐹 ∧ 𝑤 ≠ ∅ ) → ∃ 𝑢 ∈ 𝑤 ∀ 𝑓 ∈ 𝑤 ¬ 𝑓 𝑆 𝑢 ) → ( ( ( Fun 𝐹 ∧ 𝑧 ⊆ dom 𝐹 ) ∧ 𝑧 ≠ ∅ ) → ∃ 𝑢 ∈ ( 𝐹 “ 𝑧 ) ∀ 𝑓 ∈ ( 𝐹 “ 𝑧 ) ¬ 𝑓 𝑆 𝑢 ) ) ) |
27 |
9 26
|
syl |
⊢ ( Fun 𝐹 → ( ∀ 𝑤 ( ( 𝑤 ⊆ ran 𝐹 ∧ 𝑤 ≠ ∅ ) → ∃ 𝑢 ∈ 𝑤 ∀ 𝑓 ∈ 𝑤 ¬ 𝑓 𝑆 𝑢 ) → ( ( ( Fun 𝐹 ∧ 𝑧 ⊆ dom 𝐹 ) ∧ 𝑧 ≠ ∅ ) → ∃ 𝑢 ∈ ( 𝐹 “ 𝑧 ) ∀ 𝑓 ∈ ( 𝐹 “ 𝑧 ) ¬ 𝑓 𝑆 𝑢 ) ) ) |
28 |
7 27
|
syl5bi |
⊢ ( Fun 𝐹 → ( 𝑆 Fr ran 𝐹 → ( ( ( Fun 𝐹 ∧ 𝑧 ⊆ dom 𝐹 ) ∧ 𝑧 ≠ ∅ ) → ∃ 𝑢 ∈ ( 𝐹 “ 𝑧 ) ∀ 𝑓 ∈ ( 𝐹 “ 𝑧 ) ¬ 𝑓 𝑆 𝑢 ) ) ) |
29 |
28
|
com23 |
⊢ ( Fun 𝐹 → ( ( ( Fun 𝐹 ∧ 𝑧 ⊆ dom 𝐹 ) ∧ 𝑧 ≠ ∅ ) → ( 𝑆 Fr ran 𝐹 → ∃ 𝑢 ∈ ( 𝐹 “ 𝑧 ) ∀ 𝑓 ∈ ( 𝐹 “ 𝑧 ) ¬ 𝑓 𝑆 𝑢 ) ) ) |
30 |
29
|
expd |
⊢ ( Fun 𝐹 → ( ( Fun 𝐹 ∧ 𝑧 ⊆ dom 𝐹 ) → ( 𝑧 ≠ ∅ → ( 𝑆 Fr ran 𝐹 → ∃ 𝑢 ∈ ( 𝐹 “ 𝑧 ) ∀ 𝑓 ∈ ( 𝐹 “ 𝑧 ) ¬ 𝑓 𝑆 𝑢 ) ) ) ) |
31 |
30
|
anabsi5 |
⊢ ( ( Fun 𝐹 ∧ 𝑧 ⊆ dom 𝐹 ) → ( 𝑧 ≠ ∅ → ( 𝑆 Fr ran 𝐹 → ∃ 𝑢 ∈ ( 𝐹 “ 𝑧 ) ∀ 𝑓 ∈ ( 𝐹 “ 𝑧 ) ¬ 𝑓 𝑆 𝑢 ) ) ) |
32 |
31
|
impd |
⊢ ( ( Fun 𝐹 ∧ 𝑧 ⊆ dom 𝐹 ) → ( ( 𝑧 ≠ ∅ ∧ 𝑆 Fr ran 𝐹 ) → ∃ 𝑢 ∈ ( 𝐹 “ 𝑧 ) ∀ 𝑓 ∈ ( 𝐹 “ 𝑧 ) ¬ 𝑓 𝑆 𝑢 ) ) |
33 |
|
fores |
⊢ ( ( Fun 𝐹 ∧ 𝑧 ⊆ dom 𝐹 ) → ( 𝐹 ↾ 𝑧 ) : 𝑧 –onto→ ( 𝐹 “ 𝑧 ) ) |
34 |
|
vex |
⊢ 𝑣 ∈ V |
35 |
|
vex |
⊢ 𝑤 ∈ V |
36 |
|
fveq2 |
⊢ ( 𝑥 = 𝑣 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑣 ) ) |
37 |
36
|
breq1d |
⊢ ( 𝑥 = 𝑣 → ( ( 𝐹 ‘ 𝑥 ) 𝑆 ( 𝐹 ‘ 𝑦 ) ↔ ( 𝐹 ‘ 𝑣 ) 𝑆 ( 𝐹 ‘ 𝑦 ) ) ) |
38 |
|
fveq2 |
⊢ ( 𝑦 = 𝑤 → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑤 ) ) |
39 |
38
|
breq2d |
⊢ ( 𝑦 = 𝑤 → ( ( 𝐹 ‘ 𝑣 ) 𝑆 ( 𝐹 ‘ 𝑦 ) ↔ ( 𝐹 ‘ 𝑣 ) 𝑆 ( 𝐹 ‘ 𝑤 ) ) ) |
40 |
34 35 37 39 1
|
brab |
⊢ ( 𝑣 𝑅 𝑤 ↔ ( 𝐹 ‘ 𝑣 ) 𝑆 ( 𝐹 ‘ 𝑤 ) ) |
41 |
|
fvres |
⊢ ( 𝑣 ∈ 𝑧 → ( ( 𝐹 ↾ 𝑧 ) ‘ 𝑣 ) = ( 𝐹 ‘ 𝑣 ) ) |
42 |
|
fvres |
⊢ ( 𝑤 ∈ 𝑧 → ( ( 𝐹 ↾ 𝑧 ) ‘ 𝑤 ) = ( 𝐹 ‘ 𝑤 ) ) |
43 |
41 42
|
breqan12rd |
⊢ ( ( 𝑤 ∈ 𝑧 ∧ 𝑣 ∈ 𝑧 ) → ( ( ( 𝐹 ↾ 𝑧 ) ‘ 𝑣 ) 𝑆 ( ( 𝐹 ↾ 𝑧 ) ‘ 𝑤 ) ↔ ( 𝐹 ‘ 𝑣 ) 𝑆 ( 𝐹 ‘ 𝑤 ) ) ) |
44 |
40 43
|
bitr4id |
⊢ ( ( 𝑤 ∈ 𝑧 ∧ 𝑣 ∈ 𝑧 ) → ( 𝑣 𝑅 𝑤 ↔ ( ( 𝐹 ↾ 𝑧 ) ‘ 𝑣 ) 𝑆 ( ( 𝐹 ↾ 𝑧 ) ‘ 𝑤 ) ) ) |
45 |
44
|
notbid |
⊢ ( ( 𝑤 ∈ 𝑧 ∧ 𝑣 ∈ 𝑧 ) → ( ¬ 𝑣 𝑅 𝑤 ↔ ¬ ( ( 𝐹 ↾ 𝑧 ) ‘ 𝑣 ) 𝑆 ( ( 𝐹 ↾ 𝑧 ) ‘ 𝑤 ) ) ) |
46 |
45
|
ralbidva |
⊢ ( 𝑤 ∈ 𝑧 → ( ∀ 𝑣 ∈ 𝑧 ¬ 𝑣 𝑅 𝑤 ↔ ∀ 𝑣 ∈ 𝑧 ¬ ( ( 𝐹 ↾ 𝑧 ) ‘ 𝑣 ) 𝑆 ( ( 𝐹 ↾ 𝑧 ) ‘ 𝑤 ) ) ) |
47 |
46
|
rexbiia |
⊢ ( ∃ 𝑤 ∈ 𝑧 ∀ 𝑣 ∈ 𝑧 ¬ 𝑣 𝑅 𝑤 ↔ ∃ 𝑤 ∈ 𝑧 ∀ 𝑣 ∈ 𝑧 ¬ ( ( 𝐹 ↾ 𝑧 ) ‘ 𝑣 ) 𝑆 ( ( 𝐹 ↾ 𝑧 ) ‘ 𝑤 ) ) |
48 |
|
breq1 |
⊢ ( ( ( 𝐹 ↾ 𝑧 ) ‘ 𝑣 ) = 𝑓 → ( ( ( 𝐹 ↾ 𝑧 ) ‘ 𝑣 ) 𝑆 ( ( 𝐹 ↾ 𝑧 ) ‘ 𝑤 ) ↔ 𝑓 𝑆 ( ( 𝐹 ↾ 𝑧 ) ‘ 𝑤 ) ) ) |
49 |
48
|
notbid |
⊢ ( ( ( 𝐹 ↾ 𝑧 ) ‘ 𝑣 ) = 𝑓 → ( ¬ ( ( 𝐹 ↾ 𝑧 ) ‘ 𝑣 ) 𝑆 ( ( 𝐹 ↾ 𝑧 ) ‘ 𝑤 ) ↔ ¬ 𝑓 𝑆 ( ( 𝐹 ↾ 𝑧 ) ‘ 𝑤 ) ) ) |
50 |
49
|
cbvfo |
⊢ ( ( 𝐹 ↾ 𝑧 ) : 𝑧 –onto→ ( 𝐹 “ 𝑧 ) → ( ∀ 𝑣 ∈ 𝑧 ¬ ( ( 𝐹 ↾ 𝑧 ) ‘ 𝑣 ) 𝑆 ( ( 𝐹 ↾ 𝑧 ) ‘ 𝑤 ) ↔ ∀ 𝑓 ∈ ( 𝐹 “ 𝑧 ) ¬ 𝑓 𝑆 ( ( 𝐹 ↾ 𝑧 ) ‘ 𝑤 ) ) ) |
51 |
50
|
rexbidv |
⊢ ( ( 𝐹 ↾ 𝑧 ) : 𝑧 –onto→ ( 𝐹 “ 𝑧 ) → ( ∃ 𝑤 ∈ 𝑧 ∀ 𝑣 ∈ 𝑧 ¬ ( ( 𝐹 ↾ 𝑧 ) ‘ 𝑣 ) 𝑆 ( ( 𝐹 ↾ 𝑧 ) ‘ 𝑤 ) ↔ ∃ 𝑤 ∈ 𝑧 ∀ 𝑓 ∈ ( 𝐹 “ 𝑧 ) ¬ 𝑓 𝑆 ( ( 𝐹 ↾ 𝑧 ) ‘ 𝑤 ) ) ) |
52 |
|
breq2 |
⊢ ( ( ( 𝐹 ↾ 𝑧 ) ‘ 𝑤 ) = 𝑢 → ( 𝑓 𝑆 ( ( 𝐹 ↾ 𝑧 ) ‘ 𝑤 ) ↔ 𝑓 𝑆 𝑢 ) ) |
53 |
52
|
notbid |
⊢ ( ( ( 𝐹 ↾ 𝑧 ) ‘ 𝑤 ) = 𝑢 → ( ¬ 𝑓 𝑆 ( ( 𝐹 ↾ 𝑧 ) ‘ 𝑤 ) ↔ ¬ 𝑓 𝑆 𝑢 ) ) |
54 |
53
|
ralbidv |
⊢ ( ( ( 𝐹 ↾ 𝑧 ) ‘ 𝑤 ) = 𝑢 → ( ∀ 𝑓 ∈ ( 𝐹 “ 𝑧 ) ¬ 𝑓 𝑆 ( ( 𝐹 ↾ 𝑧 ) ‘ 𝑤 ) ↔ ∀ 𝑓 ∈ ( 𝐹 “ 𝑧 ) ¬ 𝑓 𝑆 𝑢 ) ) |
55 |
54
|
cbvexfo |
⊢ ( ( 𝐹 ↾ 𝑧 ) : 𝑧 –onto→ ( 𝐹 “ 𝑧 ) → ( ∃ 𝑤 ∈ 𝑧 ∀ 𝑓 ∈ ( 𝐹 “ 𝑧 ) ¬ 𝑓 𝑆 ( ( 𝐹 ↾ 𝑧 ) ‘ 𝑤 ) ↔ ∃ 𝑢 ∈ ( 𝐹 “ 𝑧 ) ∀ 𝑓 ∈ ( 𝐹 “ 𝑧 ) ¬ 𝑓 𝑆 𝑢 ) ) |
56 |
51 55
|
bitrd |
⊢ ( ( 𝐹 ↾ 𝑧 ) : 𝑧 –onto→ ( 𝐹 “ 𝑧 ) → ( ∃ 𝑤 ∈ 𝑧 ∀ 𝑣 ∈ 𝑧 ¬ ( ( 𝐹 ↾ 𝑧 ) ‘ 𝑣 ) 𝑆 ( ( 𝐹 ↾ 𝑧 ) ‘ 𝑤 ) ↔ ∃ 𝑢 ∈ ( 𝐹 “ 𝑧 ) ∀ 𝑓 ∈ ( 𝐹 “ 𝑧 ) ¬ 𝑓 𝑆 𝑢 ) ) |
57 |
47 56
|
syl5bb |
⊢ ( ( 𝐹 ↾ 𝑧 ) : 𝑧 –onto→ ( 𝐹 “ 𝑧 ) → ( ∃ 𝑤 ∈ 𝑧 ∀ 𝑣 ∈ 𝑧 ¬ 𝑣 𝑅 𝑤 ↔ ∃ 𝑢 ∈ ( 𝐹 “ 𝑧 ) ∀ 𝑓 ∈ ( 𝐹 “ 𝑧 ) ¬ 𝑓 𝑆 𝑢 ) ) |
58 |
33 57
|
syl |
⊢ ( ( Fun 𝐹 ∧ 𝑧 ⊆ dom 𝐹 ) → ( ∃ 𝑤 ∈ 𝑧 ∀ 𝑣 ∈ 𝑧 ¬ 𝑣 𝑅 𝑤 ↔ ∃ 𝑢 ∈ ( 𝐹 “ 𝑧 ) ∀ 𝑓 ∈ ( 𝐹 “ 𝑧 ) ¬ 𝑓 𝑆 𝑢 ) ) |
59 |
32 58
|
sylibrd |
⊢ ( ( Fun 𝐹 ∧ 𝑧 ⊆ dom 𝐹 ) → ( ( 𝑧 ≠ ∅ ∧ 𝑆 Fr ran 𝐹 ) → ∃ 𝑤 ∈ 𝑧 ∀ 𝑣 ∈ 𝑧 ¬ 𝑣 𝑅 𝑤 ) ) |
60 |
59
|
exp4b |
⊢ ( Fun 𝐹 → ( 𝑧 ⊆ dom 𝐹 → ( 𝑧 ≠ ∅ → ( 𝑆 Fr ran 𝐹 → ∃ 𝑤 ∈ 𝑧 ∀ 𝑣 ∈ 𝑧 ¬ 𝑣 𝑅 𝑤 ) ) ) ) |
61 |
60
|
com34 |
⊢ ( Fun 𝐹 → ( 𝑧 ⊆ dom 𝐹 → ( 𝑆 Fr ran 𝐹 → ( 𝑧 ≠ ∅ → ∃ 𝑤 ∈ 𝑧 ∀ 𝑣 ∈ 𝑧 ¬ 𝑣 𝑅 𝑤 ) ) ) ) |
62 |
61
|
com23 |
⊢ ( Fun 𝐹 → ( 𝑆 Fr ran 𝐹 → ( 𝑧 ⊆ dom 𝐹 → ( 𝑧 ≠ ∅ → ∃ 𝑤 ∈ 𝑧 ∀ 𝑣 ∈ 𝑧 ¬ 𝑣 𝑅 𝑤 ) ) ) ) |
63 |
62
|
imp4a |
⊢ ( Fun 𝐹 → ( 𝑆 Fr ran 𝐹 → ( ( 𝑧 ⊆ dom 𝐹 ∧ 𝑧 ≠ ∅ ) → ∃ 𝑤 ∈ 𝑧 ∀ 𝑣 ∈ 𝑧 ¬ 𝑣 𝑅 𝑤 ) ) ) |
64 |
63
|
alrimdv |
⊢ ( Fun 𝐹 → ( 𝑆 Fr ran 𝐹 → ∀ 𝑧 ( ( 𝑧 ⊆ dom 𝐹 ∧ 𝑧 ≠ ∅ ) → ∃ 𝑤 ∈ 𝑧 ∀ 𝑣 ∈ 𝑧 ¬ 𝑣 𝑅 𝑤 ) ) ) |
65 |
|
df-fr |
⊢ ( 𝑅 Fr dom 𝐹 ↔ ∀ 𝑧 ( ( 𝑧 ⊆ dom 𝐹 ∧ 𝑧 ≠ ∅ ) → ∃ 𝑤 ∈ 𝑧 ∀ 𝑣 ∈ 𝑧 ¬ 𝑣 𝑅 𝑤 ) ) |
66 |
64 65
|
syl6ibr |
⊢ ( Fun 𝐹 → ( 𝑆 Fr ran 𝐹 → 𝑅 Fr dom 𝐹 ) ) |
67 |
|
freq2 |
⊢ ( dom 𝐹 = 𝐴 → ( 𝑅 Fr dom 𝐹 ↔ 𝑅 Fr 𝐴 ) ) |
68 |
67
|
biimpd |
⊢ ( dom 𝐹 = 𝐴 → ( 𝑅 Fr dom 𝐹 → 𝑅 Fr 𝐴 ) ) |
69 |
66 68
|
sylan9 |
⊢ ( ( Fun 𝐹 ∧ dom 𝐹 = 𝐴 ) → ( 𝑆 Fr ran 𝐹 → 𝑅 Fr 𝐴 ) ) |
70 |
6 69
|
sylbi |
⊢ ( 𝐹 Fn 𝐴 → ( 𝑆 Fr ran 𝐹 → 𝑅 Fr 𝐴 ) ) |
71 |
5 70
|
sylan9r |
⊢ ( ( 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ) → ( 𝑆 Fr 𝐵 → 𝑅 Fr 𝐴 ) ) |
72 |
3 71
|
sylbi |
⊢ ( 𝐹 : 𝐴 –onto→ 𝐵 → ( 𝑆 Fr 𝐵 → 𝑅 Fr 𝐴 ) ) |
73 |
2 72
|
syl |
⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 → ( 𝑆 Fr 𝐵 → 𝑅 Fr 𝐴 ) ) |
74 |
|
df-f1o |
⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ↔ ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐹 : 𝐴 –onto→ 𝐵 ) ) |
75 |
|
fveq2 |
⊢ ( 𝑥 = 𝑤 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑤 ) ) |
76 |
75
|
breq1d |
⊢ ( 𝑥 = 𝑤 → ( ( 𝐹 ‘ 𝑥 ) 𝑆 ( 𝐹 ‘ 𝑦 ) ↔ ( 𝐹 ‘ 𝑤 ) 𝑆 ( 𝐹 ‘ 𝑦 ) ) ) |
77 |
|
fveq2 |
⊢ ( 𝑦 = 𝑣 → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑣 ) ) |
78 |
77
|
breq2d |
⊢ ( 𝑦 = 𝑣 → ( ( 𝐹 ‘ 𝑤 ) 𝑆 ( 𝐹 ‘ 𝑦 ) ↔ ( 𝐹 ‘ 𝑤 ) 𝑆 ( 𝐹 ‘ 𝑣 ) ) ) |
79 |
35 34 76 78 1
|
brab |
⊢ ( 𝑤 𝑅 𝑣 ↔ ( 𝐹 ‘ 𝑤 ) 𝑆 ( 𝐹 ‘ 𝑣 ) ) |
80 |
79
|
a1i |
⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ ( 𝑤 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ) ) → ( 𝑤 𝑅 𝑣 ↔ ( 𝐹 ‘ 𝑤 ) 𝑆 ( 𝐹 ‘ 𝑣 ) ) ) |
81 |
|
f1fveq |
⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ ( 𝑤 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ) ) → ( ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 𝑣 ) ↔ 𝑤 = 𝑣 ) ) |
82 |
81
|
bicomd |
⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ ( 𝑤 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ) ) → ( 𝑤 = 𝑣 ↔ ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 𝑣 ) ) ) |
83 |
40
|
a1i |
⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ ( 𝑤 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ) ) → ( 𝑣 𝑅 𝑤 ↔ ( 𝐹 ‘ 𝑣 ) 𝑆 ( 𝐹 ‘ 𝑤 ) ) ) |
84 |
80 82 83
|
3orbi123d |
⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ ( 𝑤 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ) ) → ( ( 𝑤 𝑅 𝑣 ∨ 𝑤 = 𝑣 ∨ 𝑣 𝑅 𝑤 ) ↔ ( ( 𝐹 ‘ 𝑤 ) 𝑆 ( 𝐹 ‘ 𝑣 ) ∨ ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 𝑣 ) ∨ ( 𝐹 ‘ 𝑣 ) 𝑆 ( 𝐹 ‘ 𝑤 ) ) ) ) |
85 |
84
|
2ralbidva |
⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → ( ∀ 𝑤 ∈ 𝐴 ∀ 𝑣 ∈ 𝐴 ( 𝑤 𝑅 𝑣 ∨ 𝑤 = 𝑣 ∨ 𝑣 𝑅 𝑤 ) ↔ ∀ 𝑤 ∈ 𝐴 ∀ 𝑣 ∈ 𝐴 ( ( 𝐹 ‘ 𝑤 ) 𝑆 ( 𝐹 ‘ 𝑣 ) ∨ ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 𝑣 ) ∨ ( 𝐹 ‘ 𝑣 ) 𝑆 ( 𝐹 ‘ 𝑤 ) ) ) ) |
86 |
|
breq1 |
⊢ ( ( 𝐹 ‘ 𝑤 ) = 𝑢 → ( ( 𝐹 ‘ 𝑤 ) 𝑆 ( 𝐹 ‘ 𝑣 ) ↔ 𝑢 𝑆 ( 𝐹 ‘ 𝑣 ) ) ) |
87 |
|
eqeq1 |
⊢ ( ( 𝐹 ‘ 𝑤 ) = 𝑢 → ( ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 𝑣 ) ↔ 𝑢 = ( 𝐹 ‘ 𝑣 ) ) ) |
88 |
|
breq2 |
⊢ ( ( 𝐹 ‘ 𝑤 ) = 𝑢 → ( ( 𝐹 ‘ 𝑣 ) 𝑆 ( 𝐹 ‘ 𝑤 ) ↔ ( 𝐹 ‘ 𝑣 ) 𝑆 𝑢 ) ) |
89 |
86 87 88
|
3orbi123d |
⊢ ( ( 𝐹 ‘ 𝑤 ) = 𝑢 → ( ( ( 𝐹 ‘ 𝑤 ) 𝑆 ( 𝐹 ‘ 𝑣 ) ∨ ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 𝑣 ) ∨ ( 𝐹 ‘ 𝑣 ) 𝑆 ( 𝐹 ‘ 𝑤 ) ) ↔ ( 𝑢 𝑆 ( 𝐹 ‘ 𝑣 ) ∨ 𝑢 = ( 𝐹 ‘ 𝑣 ) ∨ ( 𝐹 ‘ 𝑣 ) 𝑆 𝑢 ) ) ) |
90 |
89
|
ralbidv |
⊢ ( ( 𝐹 ‘ 𝑤 ) = 𝑢 → ( ∀ 𝑣 ∈ 𝐴 ( ( 𝐹 ‘ 𝑤 ) 𝑆 ( 𝐹 ‘ 𝑣 ) ∨ ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 𝑣 ) ∨ ( 𝐹 ‘ 𝑣 ) 𝑆 ( 𝐹 ‘ 𝑤 ) ) ↔ ∀ 𝑣 ∈ 𝐴 ( 𝑢 𝑆 ( 𝐹 ‘ 𝑣 ) ∨ 𝑢 = ( 𝐹 ‘ 𝑣 ) ∨ ( 𝐹 ‘ 𝑣 ) 𝑆 𝑢 ) ) ) |
91 |
90
|
cbvfo |
⊢ ( 𝐹 : 𝐴 –onto→ 𝐵 → ( ∀ 𝑤 ∈ 𝐴 ∀ 𝑣 ∈ 𝐴 ( ( 𝐹 ‘ 𝑤 ) 𝑆 ( 𝐹 ‘ 𝑣 ) ∨ ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 𝑣 ) ∨ ( 𝐹 ‘ 𝑣 ) 𝑆 ( 𝐹 ‘ 𝑤 ) ) ↔ ∀ 𝑢 ∈ 𝐵 ∀ 𝑣 ∈ 𝐴 ( 𝑢 𝑆 ( 𝐹 ‘ 𝑣 ) ∨ 𝑢 = ( 𝐹 ‘ 𝑣 ) ∨ ( 𝐹 ‘ 𝑣 ) 𝑆 𝑢 ) ) ) |
92 |
|
breq2 |
⊢ ( ( 𝐹 ‘ 𝑣 ) = 𝑓 → ( 𝑢 𝑆 ( 𝐹 ‘ 𝑣 ) ↔ 𝑢 𝑆 𝑓 ) ) |
93 |
|
eqeq2 |
⊢ ( ( 𝐹 ‘ 𝑣 ) = 𝑓 → ( 𝑢 = ( 𝐹 ‘ 𝑣 ) ↔ 𝑢 = 𝑓 ) ) |
94 |
|
breq1 |
⊢ ( ( 𝐹 ‘ 𝑣 ) = 𝑓 → ( ( 𝐹 ‘ 𝑣 ) 𝑆 𝑢 ↔ 𝑓 𝑆 𝑢 ) ) |
95 |
92 93 94
|
3orbi123d |
⊢ ( ( 𝐹 ‘ 𝑣 ) = 𝑓 → ( ( 𝑢 𝑆 ( 𝐹 ‘ 𝑣 ) ∨ 𝑢 = ( 𝐹 ‘ 𝑣 ) ∨ ( 𝐹 ‘ 𝑣 ) 𝑆 𝑢 ) ↔ ( 𝑢 𝑆 𝑓 ∨ 𝑢 = 𝑓 ∨ 𝑓 𝑆 𝑢 ) ) ) |
96 |
95
|
cbvfo |
⊢ ( 𝐹 : 𝐴 –onto→ 𝐵 → ( ∀ 𝑣 ∈ 𝐴 ( 𝑢 𝑆 ( 𝐹 ‘ 𝑣 ) ∨ 𝑢 = ( 𝐹 ‘ 𝑣 ) ∨ ( 𝐹 ‘ 𝑣 ) 𝑆 𝑢 ) ↔ ∀ 𝑓 ∈ 𝐵 ( 𝑢 𝑆 𝑓 ∨ 𝑢 = 𝑓 ∨ 𝑓 𝑆 𝑢 ) ) ) |
97 |
96
|
ralbidv |
⊢ ( 𝐹 : 𝐴 –onto→ 𝐵 → ( ∀ 𝑢 ∈ 𝐵 ∀ 𝑣 ∈ 𝐴 ( 𝑢 𝑆 ( 𝐹 ‘ 𝑣 ) ∨ 𝑢 = ( 𝐹 ‘ 𝑣 ) ∨ ( 𝐹 ‘ 𝑣 ) 𝑆 𝑢 ) ↔ ∀ 𝑢 ∈ 𝐵 ∀ 𝑓 ∈ 𝐵 ( 𝑢 𝑆 𝑓 ∨ 𝑢 = 𝑓 ∨ 𝑓 𝑆 𝑢 ) ) ) |
98 |
91 97
|
bitrd |
⊢ ( 𝐹 : 𝐴 –onto→ 𝐵 → ( ∀ 𝑤 ∈ 𝐴 ∀ 𝑣 ∈ 𝐴 ( ( 𝐹 ‘ 𝑤 ) 𝑆 ( 𝐹 ‘ 𝑣 ) ∨ ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 𝑣 ) ∨ ( 𝐹 ‘ 𝑣 ) 𝑆 ( 𝐹 ‘ 𝑤 ) ) ↔ ∀ 𝑢 ∈ 𝐵 ∀ 𝑓 ∈ 𝐵 ( 𝑢 𝑆 𝑓 ∨ 𝑢 = 𝑓 ∨ 𝑓 𝑆 𝑢 ) ) ) |
99 |
85 98
|
sylan9bb |
⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐹 : 𝐴 –onto→ 𝐵 ) → ( ∀ 𝑤 ∈ 𝐴 ∀ 𝑣 ∈ 𝐴 ( 𝑤 𝑅 𝑣 ∨ 𝑤 = 𝑣 ∨ 𝑣 𝑅 𝑤 ) ↔ ∀ 𝑢 ∈ 𝐵 ∀ 𝑓 ∈ 𝐵 ( 𝑢 𝑆 𝑓 ∨ 𝑢 = 𝑓 ∨ 𝑓 𝑆 𝑢 ) ) ) |
100 |
74 99
|
sylbi |
⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 → ( ∀ 𝑤 ∈ 𝐴 ∀ 𝑣 ∈ 𝐴 ( 𝑤 𝑅 𝑣 ∨ 𝑤 = 𝑣 ∨ 𝑣 𝑅 𝑤 ) ↔ ∀ 𝑢 ∈ 𝐵 ∀ 𝑓 ∈ 𝐵 ( 𝑢 𝑆 𝑓 ∨ 𝑢 = 𝑓 ∨ 𝑓 𝑆 𝑢 ) ) ) |
101 |
100
|
biimprd |
⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 → ( ∀ 𝑢 ∈ 𝐵 ∀ 𝑓 ∈ 𝐵 ( 𝑢 𝑆 𝑓 ∨ 𝑢 = 𝑓 ∨ 𝑓 𝑆 𝑢 ) → ∀ 𝑤 ∈ 𝐴 ∀ 𝑣 ∈ 𝐴 ( 𝑤 𝑅 𝑣 ∨ 𝑤 = 𝑣 ∨ 𝑣 𝑅 𝑤 ) ) ) |
102 |
73 101
|
anim12d |
⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 → ( ( 𝑆 Fr 𝐵 ∧ ∀ 𝑢 ∈ 𝐵 ∀ 𝑓 ∈ 𝐵 ( 𝑢 𝑆 𝑓 ∨ 𝑢 = 𝑓 ∨ 𝑓 𝑆 𝑢 ) ) → ( 𝑅 Fr 𝐴 ∧ ∀ 𝑤 ∈ 𝐴 ∀ 𝑣 ∈ 𝐴 ( 𝑤 𝑅 𝑣 ∨ 𝑤 = 𝑣 ∨ 𝑣 𝑅 𝑤 ) ) ) ) |
103 |
|
dfwe2 |
⊢ ( 𝑆 We 𝐵 ↔ ( 𝑆 Fr 𝐵 ∧ ∀ 𝑢 ∈ 𝐵 ∀ 𝑓 ∈ 𝐵 ( 𝑢 𝑆 𝑓 ∨ 𝑢 = 𝑓 ∨ 𝑓 𝑆 𝑢 ) ) ) |
104 |
|
dfwe2 |
⊢ ( 𝑅 We 𝐴 ↔ ( 𝑅 Fr 𝐴 ∧ ∀ 𝑤 ∈ 𝐴 ∀ 𝑣 ∈ 𝐴 ( 𝑤 𝑅 𝑣 ∨ 𝑤 = 𝑣 ∨ 𝑣 𝑅 𝑤 ) ) ) |
105 |
102 103 104
|
3imtr4g |
⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 → ( 𝑆 We 𝐵 → 𝑅 We 𝐴 ) ) |