Step |
Hyp |
Ref |
Expression |
1 |
|
elfuns.1 |
⊢ 𝐹 ∈ V |
2 |
|
elrel |
⊢ ( ( Rel 𝐹 ∧ 𝑝 ∈ 𝐹 ) → ∃ 𝑥 ∃ 𝑦 𝑝 = 〈 𝑥 , 𝑦 〉 ) |
3 |
2
|
ex |
⊢ ( Rel 𝐹 → ( 𝑝 ∈ 𝐹 → ∃ 𝑥 ∃ 𝑦 𝑝 = 〈 𝑥 , 𝑦 〉 ) ) |
4 |
|
elrel |
⊢ ( ( Rel 𝐹 ∧ 𝑞 ∈ 𝐹 ) → ∃ 𝑎 ∃ 𝑧 𝑞 = 〈 𝑎 , 𝑧 〉 ) |
5 |
4
|
ex |
⊢ ( Rel 𝐹 → ( 𝑞 ∈ 𝐹 → ∃ 𝑎 ∃ 𝑧 𝑞 = 〈 𝑎 , 𝑧 〉 ) ) |
6 |
3 5
|
anim12d |
⊢ ( Rel 𝐹 → ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) → ( ∃ 𝑥 ∃ 𝑦 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ ∃ 𝑎 ∃ 𝑧 𝑞 = 〈 𝑎 , 𝑧 〉 ) ) ) |
7 |
6
|
adantrd |
⊢ ( Rel 𝐹 → ( ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) → ( ∃ 𝑥 ∃ 𝑦 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ ∃ 𝑎 ∃ 𝑧 𝑞 = 〈 𝑎 , 𝑧 〉 ) ) ) |
8 |
7
|
pm4.71rd |
⊢ ( Rel 𝐹 → ( ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ↔ ( ( ∃ 𝑥 ∃ 𝑦 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ ∃ 𝑎 ∃ 𝑧 𝑞 = 〈 𝑎 , 𝑧 〉 ) ∧ ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ) ) ) |
9 |
|
19.41vvvv |
⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑎 ∃ 𝑧 ( ( 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ 𝑞 = 〈 𝑎 , 𝑧 〉 ) ∧ ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ) ↔ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑎 ∃ 𝑧 ( 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ 𝑞 = 〈 𝑎 , 𝑧 〉 ) ∧ ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ) ) |
10 |
|
ee4anv |
⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑎 ∃ 𝑧 ( 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ 𝑞 = 〈 𝑎 , 𝑧 〉 ) ↔ ( ∃ 𝑥 ∃ 𝑦 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ ∃ 𝑎 ∃ 𝑧 𝑞 = 〈 𝑎 , 𝑧 〉 ) ) |
11 |
10
|
anbi1i |
⊢ ( ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑎 ∃ 𝑧 ( 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ 𝑞 = 〈 𝑎 , 𝑧 〉 ) ∧ ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ) ↔ ( ( ∃ 𝑥 ∃ 𝑦 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ ∃ 𝑎 ∃ 𝑧 𝑞 = 〈 𝑎 , 𝑧 〉 ) ∧ ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ) ) |
12 |
9 11
|
bitr2i |
⊢ ( ( ( ∃ 𝑥 ∃ 𝑦 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ ∃ 𝑎 ∃ 𝑧 𝑞 = 〈 𝑎 , 𝑧 〉 ) ∧ ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ) ↔ ∃ 𝑥 ∃ 𝑦 ∃ 𝑎 ∃ 𝑧 ( ( 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ 𝑞 = 〈 𝑎 , 𝑧 〉 ) ∧ ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ) ) |
13 |
8 12
|
bitrdi |
⊢ ( Rel 𝐹 → ( ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ↔ ∃ 𝑥 ∃ 𝑦 ∃ 𝑎 ∃ 𝑧 ( ( 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ 𝑞 = 〈 𝑎 , 𝑧 〉 ) ∧ ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ) ) ) |
14 |
13
|
2exbidv |
⊢ ( Rel 𝐹 → ( ∃ 𝑝 ∃ 𝑞 ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ↔ ∃ 𝑝 ∃ 𝑞 ∃ 𝑥 ∃ 𝑦 ∃ 𝑎 ∃ 𝑧 ( ( 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ 𝑞 = 〈 𝑎 , 𝑧 〉 ) ∧ ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ) ) ) |
15 |
|
excom13 |
⊢ ( ∃ 𝑝 ∃ 𝑞 ∃ 𝑥 ∃ 𝑦 ∃ 𝑎 ∃ 𝑧 ( ( 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ 𝑞 = 〈 𝑎 , 𝑧 〉 ) ∧ ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ) ↔ ∃ 𝑥 ∃ 𝑞 ∃ 𝑝 ∃ 𝑦 ∃ 𝑎 ∃ 𝑧 ( ( 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ 𝑞 = 〈 𝑎 , 𝑧 〉 ) ∧ ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ) ) |
16 |
|
excom13 |
⊢ ( ∃ 𝑞 ∃ 𝑝 ∃ 𝑦 ∃ 𝑎 ∃ 𝑧 ( ( 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ 𝑞 = 〈 𝑎 , 𝑧 〉 ) ∧ ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ) ↔ ∃ 𝑦 ∃ 𝑝 ∃ 𝑞 ∃ 𝑎 ∃ 𝑧 ( ( 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ 𝑞 = 〈 𝑎 , 𝑧 〉 ) ∧ ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ) ) |
17 |
|
exrot4 |
⊢ ( ∃ 𝑝 ∃ 𝑞 ∃ 𝑎 ∃ 𝑧 ( ( 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ 𝑞 = 〈 𝑎 , 𝑧 〉 ) ∧ ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ) ↔ ∃ 𝑎 ∃ 𝑧 ∃ 𝑝 ∃ 𝑞 ( ( 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ 𝑞 = 〈 𝑎 , 𝑧 〉 ) ∧ ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ) ) |
18 |
|
excom |
⊢ ( ∃ 𝑎 ∃ 𝑧 ∃ 𝑝 ∃ 𝑞 ( ( 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ 𝑞 = 〈 𝑎 , 𝑧 〉 ) ∧ ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ) ↔ ∃ 𝑧 ∃ 𝑎 ∃ 𝑝 ∃ 𝑞 ( ( 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ 𝑞 = 〈 𝑎 , 𝑧 〉 ) ∧ ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ) ) |
19 |
|
df-3an |
⊢ ( ( 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ 𝑞 = 〈 𝑎 , 𝑧 〉 ∧ ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ) ↔ ( ( 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ 𝑞 = 〈 𝑎 , 𝑧 〉 ) ∧ ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ) ) |
20 |
19
|
2exbii |
⊢ ( ∃ 𝑝 ∃ 𝑞 ( 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ 𝑞 = 〈 𝑎 , 𝑧 〉 ∧ ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ) ↔ ∃ 𝑝 ∃ 𝑞 ( ( 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ 𝑞 = 〈 𝑎 , 𝑧 〉 ) ∧ ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ) ) |
21 |
|
opex |
⊢ 〈 𝑥 , 𝑦 〉 ∈ V |
22 |
|
opex |
⊢ 〈 𝑎 , 𝑧 〉 ∈ V |
23 |
|
eleq1 |
⊢ ( 𝑝 = 〈 𝑥 , 𝑦 〉 → ( 𝑝 ∈ 𝐹 ↔ 〈 𝑥 , 𝑦 〉 ∈ 𝐹 ) ) |
24 |
23
|
anbi1d |
⊢ ( 𝑝 = 〈 𝑥 , 𝑦 〉 → ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ↔ ( 〈 𝑥 , 𝑦 〉 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ) ) |
25 |
|
breq2 |
⊢ ( 𝑝 = 〈 𝑥 , 𝑦 〉 → ( 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ↔ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 〈 𝑥 , 𝑦 〉 ) ) |
26 |
24 25
|
anbi12d |
⊢ ( 𝑝 = 〈 𝑥 , 𝑦 〉 → ( ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ↔ ( ( 〈 𝑥 , 𝑦 〉 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 〈 𝑥 , 𝑦 〉 ) ) ) |
27 |
|
eleq1 |
⊢ ( 𝑞 = 〈 𝑎 , 𝑧 〉 → ( 𝑞 ∈ 𝐹 ↔ 〈 𝑎 , 𝑧 〉 ∈ 𝐹 ) ) |
28 |
27
|
anbi2d |
⊢ ( 𝑞 = 〈 𝑎 , 𝑧 〉 → ( ( 〈 𝑥 , 𝑦 〉 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ↔ ( 〈 𝑥 , 𝑦 〉 ∈ 𝐹 ∧ 〈 𝑎 , 𝑧 〉 ∈ 𝐹 ) ) ) |
29 |
|
breq1 |
⊢ ( 𝑞 = 〈 𝑎 , 𝑧 〉 → ( 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 〈 𝑥 , 𝑦 〉 ↔ 〈 𝑎 , 𝑧 〉 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 〈 𝑥 , 𝑦 〉 ) ) |
30 |
|
vex |
⊢ 𝑥 ∈ V |
31 |
|
vex |
⊢ 𝑦 ∈ V |
32 |
22 30 31
|
brtxp |
⊢ ( 〈 𝑎 , 𝑧 〉 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 〈 𝑥 , 𝑦 〉 ↔ ( 〈 𝑎 , 𝑧 〉 1st 𝑥 ∧ 〈 𝑎 , 𝑧 〉 ( ( V ∖ I ) ∘ 2nd ) 𝑦 ) ) |
33 |
|
vex |
⊢ 𝑎 ∈ V |
34 |
|
vex |
⊢ 𝑧 ∈ V |
35 |
33 34
|
br1steq |
⊢ ( 〈 𝑎 , 𝑧 〉 1st 𝑥 ↔ 𝑥 = 𝑎 ) |
36 |
|
equcom |
⊢ ( 𝑥 = 𝑎 ↔ 𝑎 = 𝑥 ) |
37 |
35 36
|
bitri |
⊢ ( 〈 𝑎 , 𝑧 〉 1st 𝑥 ↔ 𝑎 = 𝑥 ) |
38 |
22 31
|
brco |
⊢ ( 〈 𝑎 , 𝑧 〉 ( ( V ∖ I ) ∘ 2nd ) 𝑦 ↔ ∃ 𝑥 ( 〈 𝑎 , 𝑧 〉 2nd 𝑥 ∧ 𝑥 ( V ∖ I ) 𝑦 ) ) |
39 |
33 34
|
br2ndeq |
⊢ ( 〈 𝑎 , 𝑧 〉 2nd 𝑥 ↔ 𝑥 = 𝑧 ) |
40 |
39
|
anbi1i |
⊢ ( ( 〈 𝑎 , 𝑧 〉 2nd 𝑥 ∧ 𝑥 ( V ∖ I ) 𝑦 ) ↔ ( 𝑥 = 𝑧 ∧ 𝑥 ( V ∖ I ) 𝑦 ) ) |
41 |
40
|
exbii |
⊢ ( ∃ 𝑥 ( 〈 𝑎 , 𝑧 〉 2nd 𝑥 ∧ 𝑥 ( V ∖ I ) 𝑦 ) ↔ ∃ 𝑥 ( 𝑥 = 𝑧 ∧ 𝑥 ( V ∖ I ) 𝑦 ) ) |
42 |
|
breq1 |
⊢ ( 𝑥 = 𝑧 → ( 𝑥 ( V ∖ I ) 𝑦 ↔ 𝑧 ( V ∖ I ) 𝑦 ) ) |
43 |
|
brv |
⊢ 𝑧 V 𝑦 |
44 |
|
brdif |
⊢ ( 𝑧 ( V ∖ I ) 𝑦 ↔ ( 𝑧 V 𝑦 ∧ ¬ 𝑧 I 𝑦 ) ) |
45 |
43 44
|
mpbiran |
⊢ ( 𝑧 ( V ∖ I ) 𝑦 ↔ ¬ 𝑧 I 𝑦 ) |
46 |
31
|
ideq |
⊢ ( 𝑧 I 𝑦 ↔ 𝑧 = 𝑦 ) |
47 |
|
equcom |
⊢ ( 𝑧 = 𝑦 ↔ 𝑦 = 𝑧 ) |
48 |
46 47
|
bitri |
⊢ ( 𝑧 I 𝑦 ↔ 𝑦 = 𝑧 ) |
49 |
48
|
notbii |
⊢ ( ¬ 𝑧 I 𝑦 ↔ ¬ 𝑦 = 𝑧 ) |
50 |
45 49
|
bitri |
⊢ ( 𝑧 ( V ∖ I ) 𝑦 ↔ ¬ 𝑦 = 𝑧 ) |
51 |
42 50
|
bitrdi |
⊢ ( 𝑥 = 𝑧 → ( 𝑥 ( V ∖ I ) 𝑦 ↔ ¬ 𝑦 = 𝑧 ) ) |
52 |
51
|
equsexvw |
⊢ ( ∃ 𝑥 ( 𝑥 = 𝑧 ∧ 𝑥 ( V ∖ I ) 𝑦 ) ↔ ¬ 𝑦 = 𝑧 ) |
53 |
38 41 52
|
3bitri |
⊢ ( 〈 𝑎 , 𝑧 〉 ( ( V ∖ I ) ∘ 2nd ) 𝑦 ↔ ¬ 𝑦 = 𝑧 ) |
54 |
37 53
|
anbi12i |
⊢ ( ( 〈 𝑎 , 𝑧 〉 1st 𝑥 ∧ 〈 𝑎 , 𝑧 〉 ( ( V ∖ I ) ∘ 2nd ) 𝑦 ) ↔ ( 𝑎 = 𝑥 ∧ ¬ 𝑦 = 𝑧 ) ) |
55 |
32 54
|
bitri |
⊢ ( 〈 𝑎 , 𝑧 〉 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 〈 𝑥 , 𝑦 〉 ↔ ( 𝑎 = 𝑥 ∧ ¬ 𝑦 = 𝑧 ) ) |
56 |
29 55
|
bitrdi |
⊢ ( 𝑞 = 〈 𝑎 , 𝑧 〉 → ( 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 〈 𝑥 , 𝑦 〉 ↔ ( 𝑎 = 𝑥 ∧ ¬ 𝑦 = 𝑧 ) ) ) |
57 |
28 56
|
anbi12d |
⊢ ( 𝑞 = 〈 𝑎 , 𝑧 〉 → ( ( ( 〈 𝑥 , 𝑦 〉 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 〈 𝑥 , 𝑦 〉 ) ↔ ( ( 〈 𝑥 , 𝑦 〉 ∈ 𝐹 ∧ 〈 𝑎 , 𝑧 〉 ∈ 𝐹 ) ∧ ( 𝑎 = 𝑥 ∧ ¬ 𝑦 = 𝑧 ) ) ) ) |
58 |
|
an12 |
⊢ ( ( ( 〈 𝑥 , 𝑦 〉 ∈ 𝐹 ∧ 〈 𝑎 , 𝑧 〉 ∈ 𝐹 ) ∧ ( 𝑎 = 𝑥 ∧ ¬ 𝑦 = 𝑧 ) ) ↔ ( 𝑎 = 𝑥 ∧ ( ( 〈 𝑥 , 𝑦 〉 ∈ 𝐹 ∧ 〈 𝑎 , 𝑧 〉 ∈ 𝐹 ) ∧ ¬ 𝑦 = 𝑧 ) ) ) |
59 |
57 58
|
bitrdi |
⊢ ( 𝑞 = 〈 𝑎 , 𝑧 〉 → ( ( ( 〈 𝑥 , 𝑦 〉 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 〈 𝑥 , 𝑦 〉 ) ↔ ( 𝑎 = 𝑥 ∧ ( ( 〈 𝑥 , 𝑦 〉 ∈ 𝐹 ∧ 〈 𝑎 , 𝑧 〉 ∈ 𝐹 ) ∧ ¬ 𝑦 = 𝑧 ) ) ) ) |
60 |
21 22 26 59
|
ceqsex2v |
⊢ ( ∃ 𝑝 ∃ 𝑞 ( 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ 𝑞 = 〈 𝑎 , 𝑧 〉 ∧ ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ) ↔ ( 𝑎 = 𝑥 ∧ ( ( 〈 𝑥 , 𝑦 〉 ∈ 𝐹 ∧ 〈 𝑎 , 𝑧 〉 ∈ 𝐹 ) ∧ ¬ 𝑦 = 𝑧 ) ) ) |
61 |
20 60
|
bitr3i |
⊢ ( ∃ 𝑝 ∃ 𝑞 ( ( 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ 𝑞 = 〈 𝑎 , 𝑧 〉 ) ∧ ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ) ↔ ( 𝑎 = 𝑥 ∧ ( ( 〈 𝑥 , 𝑦 〉 ∈ 𝐹 ∧ 〈 𝑎 , 𝑧 〉 ∈ 𝐹 ) ∧ ¬ 𝑦 = 𝑧 ) ) ) |
62 |
61
|
exbii |
⊢ ( ∃ 𝑎 ∃ 𝑝 ∃ 𝑞 ( ( 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ 𝑞 = 〈 𝑎 , 𝑧 〉 ) ∧ ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ) ↔ ∃ 𝑎 ( 𝑎 = 𝑥 ∧ ( ( 〈 𝑥 , 𝑦 〉 ∈ 𝐹 ∧ 〈 𝑎 , 𝑧 〉 ∈ 𝐹 ) ∧ ¬ 𝑦 = 𝑧 ) ) ) |
63 |
|
opeq1 |
⊢ ( 𝑎 = 𝑥 → 〈 𝑎 , 𝑧 〉 = 〈 𝑥 , 𝑧 〉 ) |
64 |
63
|
eleq1d |
⊢ ( 𝑎 = 𝑥 → ( 〈 𝑎 , 𝑧 〉 ∈ 𝐹 ↔ 〈 𝑥 , 𝑧 〉 ∈ 𝐹 ) ) |
65 |
64
|
anbi2d |
⊢ ( 𝑎 = 𝑥 → ( ( 〈 𝑥 , 𝑦 〉 ∈ 𝐹 ∧ 〈 𝑎 , 𝑧 〉 ∈ 𝐹 ) ↔ ( 〈 𝑥 , 𝑦 〉 ∈ 𝐹 ∧ 〈 𝑥 , 𝑧 〉 ∈ 𝐹 ) ) ) |
66 |
65
|
anbi1d |
⊢ ( 𝑎 = 𝑥 → ( ( ( 〈 𝑥 , 𝑦 〉 ∈ 𝐹 ∧ 〈 𝑎 , 𝑧 〉 ∈ 𝐹 ) ∧ ¬ 𝑦 = 𝑧 ) ↔ ( ( 〈 𝑥 , 𝑦 〉 ∈ 𝐹 ∧ 〈 𝑥 , 𝑧 〉 ∈ 𝐹 ) ∧ ¬ 𝑦 = 𝑧 ) ) ) |
67 |
66
|
equsexvw |
⊢ ( ∃ 𝑎 ( 𝑎 = 𝑥 ∧ ( ( 〈 𝑥 , 𝑦 〉 ∈ 𝐹 ∧ 〈 𝑎 , 𝑧 〉 ∈ 𝐹 ) ∧ ¬ 𝑦 = 𝑧 ) ) ↔ ( ( 〈 𝑥 , 𝑦 〉 ∈ 𝐹 ∧ 〈 𝑥 , 𝑧 〉 ∈ 𝐹 ) ∧ ¬ 𝑦 = 𝑧 ) ) |
68 |
62 67
|
bitri |
⊢ ( ∃ 𝑎 ∃ 𝑝 ∃ 𝑞 ( ( 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ 𝑞 = 〈 𝑎 , 𝑧 〉 ) ∧ ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ) ↔ ( ( 〈 𝑥 , 𝑦 〉 ∈ 𝐹 ∧ 〈 𝑥 , 𝑧 〉 ∈ 𝐹 ) ∧ ¬ 𝑦 = 𝑧 ) ) |
69 |
68
|
exbii |
⊢ ( ∃ 𝑧 ∃ 𝑎 ∃ 𝑝 ∃ 𝑞 ( ( 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ 𝑞 = 〈 𝑎 , 𝑧 〉 ) ∧ ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ) ↔ ∃ 𝑧 ( ( 〈 𝑥 , 𝑦 〉 ∈ 𝐹 ∧ 〈 𝑥 , 𝑧 〉 ∈ 𝐹 ) ∧ ¬ 𝑦 = 𝑧 ) ) |
70 |
|
exanali |
⊢ ( ∃ 𝑧 ( ( 〈 𝑥 , 𝑦 〉 ∈ 𝐹 ∧ 〈 𝑥 , 𝑧 〉 ∈ 𝐹 ) ∧ ¬ 𝑦 = 𝑧 ) ↔ ¬ ∀ 𝑧 ( ( 〈 𝑥 , 𝑦 〉 ∈ 𝐹 ∧ 〈 𝑥 , 𝑧 〉 ∈ 𝐹 ) → 𝑦 = 𝑧 ) ) |
71 |
69 70
|
bitri |
⊢ ( ∃ 𝑧 ∃ 𝑎 ∃ 𝑝 ∃ 𝑞 ( ( 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ 𝑞 = 〈 𝑎 , 𝑧 〉 ) ∧ ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ) ↔ ¬ ∀ 𝑧 ( ( 〈 𝑥 , 𝑦 〉 ∈ 𝐹 ∧ 〈 𝑥 , 𝑧 〉 ∈ 𝐹 ) → 𝑦 = 𝑧 ) ) |
72 |
17 18 71
|
3bitri |
⊢ ( ∃ 𝑝 ∃ 𝑞 ∃ 𝑎 ∃ 𝑧 ( ( 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ 𝑞 = 〈 𝑎 , 𝑧 〉 ) ∧ ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ) ↔ ¬ ∀ 𝑧 ( ( 〈 𝑥 , 𝑦 〉 ∈ 𝐹 ∧ 〈 𝑥 , 𝑧 〉 ∈ 𝐹 ) → 𝑦 = 𝑧 ) ) |
73 |
72
|
exbii |
⊢ ( ∃ 𝑦 ∃ 𝑝 ∃ 𝑞 ∃ 𝑎 ∃ 𝑧 ( ( 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ 𝑞 = 〈 𝑎 , 𝑧 〉 ) ∧ ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ) ↔ ∃ 𝑦 ¬ ∀ 𝑧 ( ( 〈 𝑥 , 𝑦 〉 ∈ 𝐹 ∧ 〈 𝑥 , 𝑧 〉 ∈ 𝐹 ) → 𝑦 = 𝑧 ) ) |
74 |
|
exnal |
⊢ ( ∃ 𝑦 ¬ ∀ 𝑧 ( ( 〈 𝑥 , 𝑦 〉 ∈ 𝐹 ∧ 〈 𝑥 , 𝑧 〉 ∈ 𝐹 ) → 𝑦 = 𝑧 ) ↔ ¬ ∀ 𝑦 ∀ 𝑧 ( ( 〈 𝑥 , 𝑦 〉 ∈ 𝐹 ∧ 〈 𝑥 , 𝑧 〉 ∈ 𝐹 ) → 𝑦 = 𝑧 ) ) |
75 |
16 73 74
|
3bitri |
⊢ ( ∃ 𝑞 ∃ 𝑝 ∃ 𝑦 ∃ 𝑎 ∃ 𝑧 ( ( 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ 𝑞 = 〈 𝑎 , 𝑧 〉 ) ∧ ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ) ↔ ¬ ∀ 𝑦 ∀ 𝑧 ( ( 〈 𝑥 , 𝑦 〉 ∈ 𝐹 ∧ 〈 𝑥 , 𝑧 〉 ∈ 𝐹 ) → 𝑦 = 𝑧 ) ) |
76 |
75
|
exbii |
⊢ ( ∃ 𝑥 ∃ 𝑞 ∃ 𝑝 ∃ 𝑦 ∃ 𝑎 ∃ 𝑧 ( ( 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ 𝑞 = 〈 𝑎 , 𝑧 〉 ) ∧ ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ) ↔ ∃ 𝑥 ¬ ∀ 𝑦 ∀ 𝑧 ( ( 〈 𝑥 , 𝑦 〉 ∈ 𝐹 ∧ 〈 𝑥 , 𝑧 〉 ∈ 𝐹 ) → 𝑦 = 𝑧 ) ) |
77 |
|
exnal |
⊢ ( ∃ 𝑥 ¬ ∀ 𝑦 ∀ 𝑧 ( ( 〈 𝑥 , 𝑦 〉 ∈ 𝐹 ∧ 〈 𝑥 , 𝑧 〉 ∈ 𝐹 ) → 𝑦 = 𝑧 ) ↔ ¬ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 〈 𝑥 , 𝑦 〉 ∈ 𝐹 ∧ 〈 𝑥 , 𝑧 〉 ∈ 𝐹 ) → 𝑦 = 𝑧 ) ) |
78 |
15 76 77
|
3bitri |
⊢ ( ∃ 𝑝 ∃ 𝑞 ∃ 𝑥 ∃ 𝑦 ∃ 𝑎 ∃ 𝑧 ( ( 𝑝 = 〈 𝑥 , 𝑦 〉 ∧ 𝑞 = 〈 𝑎 , 𝑧 〉 ) ∧ ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ) ↔ ¬ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 〈 𝑥 , 𝑦 〉 ∈ 𝐹 ∧ 〈 𝑥 , 𝑧 〉 ∈ 𝐹 ) → 𝑦 = 𝑧 ) ) |
79 |
14 78
|
bitrdi |
⊢ ( Rel 𝐹 → ( ∃ 𝑝 ∃ 𝑞 ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ↔ ¬ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 〈 𝑥 , 𝑦 〉 ∈ 𝐹 ∧ 〈 𝑥 , 𝑧 〉 ∈ 𝐹 ) → 𝑦 = 𝑧 ) ) ) |
80 |
79
|
con2bid |
⊢ ( Rel 𝐹 → ( ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 〈 𝑥 , 𝑦 〉 ∈ 𝐹 ∧ 〈 𝑥 , 𝑧 〉 ∈ 𝐹 ) → 𝑦 = 𝑧 ) ↔ ¬ ∃ 𝑝 ∃ 𝑞 ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ) ) |
81 |
80
|
pm5.32i |
⊢ ( ( Rel 𝐹 ∧ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 〈 𝑥 , 𝑦 〉 ∈ 𝐹 ∧ 〈 𝑥 , 𝑧 〉 ∈ 𝐹 ) → 𝑦 = 𝑧 ) ) ↔ ( Rel 𝐹 ∧ ¬ ∃ 𝑝 ∃ 𝑞 ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ) ) |
82 |
|
dffun4 |
⊢ ( Fun 𝐹 ↔ ( Rel 𝐹 ∧ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 〈 𝑥 , 𝑦 〉 ∈ 𝐹 ∧ 〈 𝑥 , 𝑧 〉 ∈ 𝐹 ) → 𝑦 = 𝑧 ) ) ) |
83 |
|
df-funs |
⊢ Funs = ( 𝒫 ( V × V ) ∖ Fix ( E ∘ ( ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) ∘ ◡ E ) ) ) |
84 |
83
|
eleq2i |
⊢ ( 𝐹 ∈ Funs ↔ 𝐹 ∈ ( 𝒫 ( V × V ) ∖ Fix ( E ∘ ( ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) ∘ ◡ E ) ) ) ) |
85 |
|
eldif |
⊢ ( 𝐹 ∈ ( 𝒫 ( V × V ) ∖ Fix ( E ∘ ( ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) ∘ ◡ E ) ) ) ↔ ( 𝐹 ∈ 𝒫 ( V × V ) ∧ ¬ 𝐹 ∈ Fix ( E ∘ ( ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) ∘ ◡ E ) ) ) ) |
86 |
1
|
elpw |
⊢ ( 𝐹 ∈ 𝒫 ( V × V ) ↔ 𝐹 ⊆ ( V × V ) ) |
87 |
|
df-rel |
⊢ ( Rel 𝐹 ↔ 𝐹 ⊆ ( V × V ) ) |
88 |
86 87
|
bitr4i |
⊢ ( 𝐹 ∈ 𝒫 ( V × V ) ↔ Rel 𝐹 ) |
89 |
1
|
elfix |
⊢ ( 𝐹 ∈ Fix ( E ∘ ( ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) ∘ ◡ E ) ) ↔ 𝐹 ( E ∘ ( ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) ∘ ◡ E ) ) 𝐹 ) |
90 |
1 1
|
coep |
⊢ ( 𝐹 ( E ∘ ( ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) ∘ ◡ E ) ) 𝐹 ↔ ∃ 𝑝 ∈ 𝐹 𝐹 ( ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) ∘ ◡ E ) 𝑝 ) |
91 |
|
vex |
⊢ 𝑝 ∈ V |
92 |
1 91
|
coepr |
⊢ ( 𝐹 ( ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) ∘ ◡ E ) 𝑝 ↔ ∃ 𝑞 ∈ 𝐹 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) |
93 |
92
|
rexbii |
⊢ ( ∃ 𝑝 ∈ 𝐹 𝐹 ( ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) ∘ ◡ E ) 𝑝 ↔ ∃ 𝑝 ∈ 𝐹 ∃ 𝑞 ∈ 𝐹 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) |
94 |
90 93
|
bitri |
⊢ ( 𝐹 ( E ∘ ( ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) ∘ ◡ E ) ) 𝐹 ↔ ∃ 𝑝 ∈ 𝐹 ∃ 𝑞 ∈ 𝐹 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) |
95 |
|
r2ex |
⊢ ( ∃ 𝑝 ∈ 𝐹 ∃ 𝑞 ∈ 𝐹 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ↔ ∃ 𝑝 ∃ 𝑞 ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ) |
96 |
89 94 95
|
3bitri |
⊢ ( 𝐹 ∈ Fix ( E ∘ ( ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) ∘ ◡ E ) ) ↔ ∃ 𝑝 ∃ 𝑞 ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ) |
97 |
96
|
notbii |
⊢ ( ¬ 𝐹 ∈ Fix ( E ∘ ( ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) ∘ ◡ E ) ) ↔ ¬ ∃ 𝑝 ∃ 𝑞 ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ) |
98 |
88 97
|
anbi12i |
⊢ ( ( 𝐹 ∈ 𝒫 ( V × V ) ∧ ¬ 𝐹 ∈ Fix ( E ∘ ( ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) ∘ ◡ E ) ) ) ↔ ( Rel 𝐹 ∧ ¬ ∃ 𝑝 ∃ 𝑞 ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ) ) |
99 |
84 85 98
|
3bitri |
⊢ ( 𝐹 ∈ Funs ↔ ( Rel 𝐹 ∧ ¬ ∃ 𝑝 ∃ 𝑞 ( ( 𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹 ) ∧ 𝑞 ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) 𝑝 ) ) ) |
100 |
81 82 99
|
3bitr4ri |
⊢ ( 𝐹 ∈ Funs ↔ Fun 𝐹 ) |