Step |
Hyp |
Ref |
Expression |
1 |
|
sprsymrelf.p |
⊢ 𝑃 = 𝒫 ( Pairs ‘ 𝑉 ) |
2 |
|
sprsymrelf.r |
⊢ 𝑅 = { 𝑟 ∈ 𝒫 ( 𝑉 × 𝑉 ) ∣ ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( 𝑥 𝑟 𝑦 ↔ 𝑦 𝑟 𝑥 ) } |
3 |
|
sprsymrelf.f |
⊢ 𝐹 = ( 𝑝 ∈ 𝑃 ↦ { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑐 ∈ 𝑝 𝑐 = { 𝑥 , 𝑦 } } ) |
4 |
1 2 3
|
sprsymrelf |
⊢ 𝐹 : 𝑃 ⟶ 𝑅 |
5 |
4
|
a1i |
⊢ ( 𝑉 ∈ 𝑊 → 𝐹 : 𝑃 ⟶ 𝑅 ) |
6 |
|
breq |
⊢ ( 𝑟 = 𝑡 → ( 𝑥 𝑟 𝑦 ↔ 𝑥 𝑡 𝑦 ) ) |
7 |
|
breq |
⊢ ( 𝑟 = 𝑡 → ( 𝑦 𝑟 𝑥 ↔ 𝑦 𝑡 𝑥 ) ) |
8 |
6 7
|
bibi12d |
⊢ ( 𝑟 = 𝑡 → ( ( 𝑥 𝑟 𝑦 ↔ 𝑦 𝑟 𝑥 ) ↔ ( 𝑥 𝑡 𝑦 ↔ 𝑦 𝑡 𝑥 ) ) ) |
9 |
8
|
2ralbidv |
⊢ ( 𝑟 = 𝑡 → ( ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( 𝑥 𝑟 𝑦 ↔ 𝑦 𝑟 𝑥 ) ↔ ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( 𝑥 𝑡 𝑦 ↔ 𝑦 𝑡 𝑥 ) ) ) |
10 |
9 2
|
elrab2 |
⊢ ( 𝑡 ∈ 𝑅 ↔ ( 𝑡 ∈ 𝒫 ( 𝑉 × 𝑉 ) ∧ ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( 𝑥 𝑡 𝑦 ↔ 𝑦 𝑡 𝑥 ) ) ) |
11 |
|
eqid |
⊢ { 𝑞 ∈ ( Pairs ‘ 𝑉 ) ∣ ∀ 𝑎 ∈ 𝑉 ∀ 𝑏 ∈ 𝑉 ( 𝑞 = { 𝑎 , 𝑏 } → 𝑎 𝑡 𝑏 ) } = { 𝑞 ∈ ( Pairs ‘ 𝑉 ) ∣ ∀ 𝑎 ∈ 𝑉 ∀ 𝑏 ∈ 𝑉 ( 𝑞 = { 𝑎 , 𝑏 } → 𝑎 𝑡 𝑏 ) } |
12 |
11
|
sprsymrelfolem1 |
⊢ { 𝑞 ∈ ( Pairs ‘ 𝑉 ) ∣ ∀ 𝑎 ∈ 𝑉 ∀ 𝑏 ∈ 𝑉 ( 𝑞 = { 𝑎 , 𝑏 } → 𝑎 𝑡 𝑏 ) } ∈ 𝒫 ( Pairs ‘ 𝑉 ) |
13 |
12 1
|
eleqtrri |
⊢ { 𝑞 ∈ ( Pairs ‘ 𝑉 ) ∣ ∀ 𝑎 ∈ 𝑉 ∀ 𝑏 ∈ 𝑉 ( 𝑞 = { 𝑎 , 𝑏 } → 𝑎 𝑡 𝑏 ) } ∈ 𝑃 |
14 |
13
|
a1i |
⊢ ( ( ( 𝑡 ∈ 𝒫 ( 𝑉 × 𝑉 ) ∧ ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( 𝑥 𝑡 𝑦 ↔ 𝑦 𝑡 𝑥 ) ) ∧ 𝑉 ∈ 𝑊 ) → { 𝑞 ∈ ( Pairs ‘ 𝑉 ) ∣ ∀ 𝑎 ∈ 𝑉 ∀ 𝑏 ∈ 𝑉 ( 𝑞 = { 𝑎 , 𝑏 } → 𝑎 𝑡 𝑏 ) } ∈ 𝑃 ) |
15 |
|
rexeq |
⊢ ( 𝑓 = { 𝑞 ∈ ( Pairs ‘ 𝑉 ) ∣ ∀ 𝑎 ∈ 𝑉 ∀ 𝑏 ∈ 𝑉 ( 𝑞 = { 𝑎 , 𝑏 } → 𝑎 𝑡 𝑏 ) } → ( ∃ 𝑐 ∈ 𝑓 𝑐 = { 𝑥 , 𝑦 } ↔ ∃ 𝑐 ∈ { 𝑞 ∈ ( Pairs ‘ 𝑉 ) ∣ ∀ 𝑎 ∈ 𝑉 ∀ 𝑏 ∈ 𝑉 ( 𝑞 = { 𝑎 , 𝑏 } → 𝑎 𝑡 𝑏 ) } 𝑐 = { 𝑥 , 𝑦 } ) ) |
16 |
15
|
opabbidv |
⊢ ( 𝑓 = { 𝑞 ∈ ( Pairs ‘ 𝑉 ) ∣ ∀ 𝑎 ∈ 𝑉 ∀ 𝑏 ∈ 𝑉 ( 𝑞 = { 𝑎 , 𝑏 } → 𝑎 𝑡 𝑏 ) } → { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑐 ∈ 𝑓 𝑐 = { 𝑥 , 𝑦 } } = { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑐 ∈ { 𝑞 ∈ ( Pairs ‘ 𝑉 ) ∣ ∀ 𝑎 ∈ 𝑉 ∀ 𝑏 ∈ 𝑉 ( 𝑞 = { 𝑎 , 𝑏 } → 𝑎 𝑡 𝑏 ) } 𝑐 = { 𝑥 , 𝑦 } } ) |
17 |
16
|
eqeq2d |
⊢ ( 𝑓 = { 𝑞 ∈ ( Pairs ‘ 𝑉 ) ∣ ∀ 𝑎 ∈ 𝑉 ∀ 𝑏 ∈ 𝑉 ( 𝑞 = { 𝑎 , 𝑏 } → 𝑎 𝑡 𝑏 ) } → ( 𝑡 = { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑐 ∈ 𝑓 𝑐 = { 𝑥 , 𝑦 } } ↔ 𝑡 = { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑐 ∈ { 𝑞 ∈ ( Pairs ‘ 𝑉 ) ∣ ∀ 𝑎 ∈ 𝑉 ∀ 𝑏 ∈ 𝑉 ( 𝑞 = { 𝑎 , 𝑏 } → 𝑎 𝑡 𝑏 ) } 𝑐 = { 𝑥 , 𝑦 } } ) ) |
18 |
17
|
adantl |
⊢ ( ( ( ( 𝑡 ∈ 𝒫 ( 𝑉 × 𝑉 ) ∧ ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( 𝑥 𝑡 𝑦 ↔ 𝑦 𝑡 𝑥 ) ) ∧ 𝑉 ∈ 𝑊 ) ∧ 𝑓 = { 𝑞 ∈ ( Pairs ‘ 𝑉 ) ∣ ∀ 𝑎 ∈ 𝑉 ∀ 𝑏 ∈ 𝑉 ( 𝑞 = { 𝑎 , 𝑏 } → 𝑎 𝑡 𝑏 ) } ) → ( 𝑡 = { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑐 ∈ 𝑓 𝑐 = { 𝑥 , 𝑦 } } ↔ 𝑡 = { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑐 ∈ { 𝑞 ∈ ( Pairs ‘ 𝑉 ) ∣ ∀ 𝑎 ∈ 𝑉 ∀ 𝑏 ∈ 𝑉 ( 𝑞 = { 𝑎 , 𝑏 } → 𝑎 𝑡 𝑏 ) } 𝑐 = { 𝑥 , 𝑦 } } ) ) |
19 |
|
velpw |
⊢ ( 𝑡 ∈ 𝒫 ( 𝑉 × 𝑉 ) ↔ 𝑡 ⊆ ( 𝑉 × 𝑉 ) ) |
20 |
|
xpss |
⊢ ( 𝑉 × 𝑉 ) ⊆ ( V × V ) |
21 |
|
sstr2 |
⊢ ( 𝑡 ⊆ ( 𝑉 × 𝑉 ) → ( ( 𝑉 × 𝑉 ) ⊆ ( V × V ) → 𝑡 ⊆ ( V × V ) ) ) |
22 |
20 21
|
mpi |
⊢ ( 𝑡 ⊆ ( 𝑉 × 𝑉 ) → 𝑡 ⊆ ( V × V ) ) |
23 |
|
df-rel |
⊢ ( Rel 𝑡 ↔ 𝑡 ⊆ ( V × V ) ) |
24 |
22 23
|
sylibr |
⊢ ( 𝑡 ⊆ ( 𝑉 × 𝑉 ) → Rel 𝑡 ) |
25 |
24
|
adantl |
⊢ ( ( 𝑉 ∈ 𝑊 ∧ 𝑡 ⊆ ( 𝑉 × 𝑉 ) ) → Rel 𝑡 ) |
26 |
|
dfrel4v |
⊢ ( Rel 𝑡 ↔ 𝑡 = { 〈 𝑥 , 𝑦 〉 ∣ 𝑥 𝑡 𝑦 } ) |
27 |
|
nfv |
⊢ Ⅎ 𝑥 ( 𝑉 ∈ 𝑊 ∧ 𝑡 ⊆ ( 𝑉 × 𝑉 ) ) |
28 |
|
nfra1 |
⊢ Ⅎ 𝑥 ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( 𝑥 𝑡 𝑦 ↔ 𝑦 𝑡 𝑥 ) |
29 |
27 28
|
nfan |
⊢ Ⅎ 𝑥 ( ( 𝑉 ∈ 𝑊 ∧ 𝑡 ⊆ ( 𝑉 × 𝑉 ) ) ∧ ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( 𝑥 𝑡 𝑦 ↔ 𝑦 𝑡 𝑥 ) ) |
30 |
|
nfv |
⊢ Ⅎ 𝑦 ( 𝑉 ∈ 𝑊 ∧ 𝑡 ⊆ ( 𝑉 × 𝑉 ) ) |
31 |
|
nfra2w |
⊢ Ⅎ 𝑦 ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( 𝑥 𝑡 𝑦 ↔ 𝑦 𝑡 𝑥 ) |
32 |
30 31
|
nfan |
⊢ Ⅎ 𝑦 ( ( 𝑉 ∈ 𝑊 ∧ 𝑡 ⊆ ( 𝑉 × 𝑉 ) ) ∧ ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( 𝑥 𝑡 𝑦 ↔ 𝑦 𝑡 𝑥 ) ) |
33 |
11
|
sprsymrelfolem2 |
⊢ ( ( 𝑉 ∈ 𝑊 ∧ 𝑡 ⊆ ( 𝑉 × 𝑉 ) ∧ ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( 𝑥 𝑡 𝑦 ↔ 𝑦 𝑡 𝑥 ) ) → ( 𝑥 𝑡 𝑦 ↔ ∃ 𝑐 ∈ { 𝑞 ∈ ( Pairs ‘ 𝑉 ) ∣ ∀ 𝑎 ∈ 𝑉 ∀ 𝑏 ∈ 𝑉 ( 𝑞 = { 𝑎 , 𝑏 } → 𝑎 𝑡 𝑏 ) } 𝑐 = { 𝑥 , 𝑦 } ) ) |
34 |
33
|
3expa |
⊢ ( ( ( 𝑉 ∈ 𝑊 ∧ 𝑡 ⊆ ( 𝑉 × 𝑉 ) ) ∧ ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( 𝑥 𝑡 𝑦 ↔ 𝑦 𝑡 𝑥 ) ) → ( 𝑥 𝑡 𝑦 ↔ ∃ 𝑐 ∈ { 𝑞 ∈ ( Pairs ‘ 𝑉 ) ∣ ∀ 𝑎 ∈ 𝑉 ∀ 𝑏 ∈ 𝑉 ( 𝑞 = { 𝑎 , 𝑏 } → 𝑎 𝑡 𝑏 ) } 𝑐 = { 𝑥 , 𝑦 } ) ) |
35 |
29 32 34
|
opabbid |
⊢ ( ( ( 𝑉 ∈ 𝑊 ∧ 𝑡 ⊆ ( 𝑉 × 𝑉 ) ) ∧ ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( 𝑥 𝑡 𝑦 ↔ 𝑦 𝑡 𝑥 ) ) → { 〈 𝑥 , 𝑦 〉 ∣ 𝑥 𝑡 𝑦 } = { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑐 ∈ { 𝑞 ∈ ( Pairs ‘ 𝑉 ) ∣ ∀ 𝑎 ∈ 𝑉 ∀ 𝑏 ∈ 𝑉 ( 𝑞 = { 𝑎 , 𝑏 } → 𝑎 𝑡 𝑏 ) } 𝑐 = { 𝑥 , 𝑦 } } ) |
36 |
35
|
eqeq2d |
⊢ ( ( ( 𝑉 ∈ 𝑊 ∧ 𝑡 ⊆ ( 𝑉 × 𝑉 ) ) ∧ ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( 𝑥 𝑡 𝑦 ↔ 𝑦 𝑡 𝑥 ) ) → ( 𝑡 = { 〈 𝑥 , 𝑦 〉 ∣ 𝑥 𝑡 𝑦 } ↔ 𝑡 = { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑐 ∈ { 𝑞 ∈ ( Pairs ‘ 𝑉 ) ∣ ∀ 𝑎 ∈ 𝑉 ∀ 𝑏 ∈ 𝑉 ( 𝑞 = { 𝑎 , 𝑏 } → 𝑎 𝑡 𝑏 ) } 𝑐 = { 𝑥 , 𝑦 } } ) ) |
37 |
36
|
biimpd |
⊢ ( ( ( 𝑉 ∈ 𝑊 ∧ 𝑡 ⊆ ( 𝑉 × 𝑉 ) ) ∧ ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( 𝑥 𝑡 𝑦 ↔ 𝑦 𝑡 𝑥 ) ) → ( 𝑡 = { 〈 𝑥 , 𝑦 〉 ∣ 𝑥 𝑡 𝑦 } → 𝑡 = { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑐 ∈ { 𝑞 ∈ ( Pairs ‘ 𝑉 ) ∣ ∀ 𝑎 ∈ 𝑉 ∀ 𝑏 ∈ 𝑉 ( 𝑞 = { 𝑎 , 𝑏 } → 𝑎 𝑡 𝑏 ) } 𝑐 = { 𝑥 , 𝑦 } } ) ) |
38 |
37
|
ex |
⊢ ( ( 𝑉 ∈ 𝑊 ∧ 𝑡 ⊆ ( 𝑉 × 𝑉 ) ) → ( ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( 𝑥 𝑡 𝑦 ↔ 𝑦 𝑡 𝑥 ) → ( 𝑡 = { 〈 𝑥 , 𝑦 〉 ∣ 𝑥 𝑡 𝑦 } → 𝑡 = { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑐 ∈ { 𝑞 ∈ ( Pairs ‘ 𝑉 ) ∣ ∀ 𝑎 ∈ 𝑉 ∀ 𝑏 ∈ 𝑉 ( 𝑞 = { 𝑎 , 𝑏 } → 𝑎 𝑡 𝑏 ) } 𝑐 = { 𝑥 , 𝑦 } } ) ) ) |
39 |
38
|
com23 |
⊢ ( ( 𝑉 ∈ 𝑊 ∧ 𝑡 ⊆ ( 𝑉 × 𝑉 ) ) → ( 𝑡 = { 〈 𝑥 , 𝑦 〉 ∣ 𝑥 𝑡 𝑦 } → ( ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( 𝑥 𝑡 𝑦 ↔ 𝑦 𝑡 𝑥 ) → 𝑡 = { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑐 ∈ { 𝑞 ∈ ( Pairs ‘ 𝑉 ) ∣ ∀ 𝑎 ∈ 𝑉 ∀ 𝑏 ∈ 𝑉 ( 𝑞 = { 𝑎 , 𝑏 } → 𝑎 𝑡 𝑏 ) } 𝑐 = { 𝑥 , 𝑦 } } ) ) ) |
40 |
26 39
|
syl5bi |
⊢ ( ( 𝑉 ∈ 𝑊 ∧ 𝑡 ⊆ ( 𝑉 × 𝑉 ) ) → ( Rel 𝑡 → ( ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( 𝑥 𝑡 𝑦 ↔ 𝑦 𝑡 𝑥 ) → 𝑡 = { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑐 ∈ { 𝑞 ∈ ( Pairs ‘ 𝑉 ) ∣ ∀ 𝑎 ∈ 𝑉 ∀ 𝑏 ∈ 𝑉 ( 𝑞 = { 𝑎 , 𝑏 } → 𝑎 𝑡 𝑏 ) } 𝑐 = { 𝑥 , 𝑦 } } ) ) ) |
41 |
25 40
|
mpd |
⊢ ( ( 𝑉 ∈ 𝑊 ∧ 𝑡 ⊆ ( 𝑉 × 𝑉 ) ) → ( ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( 𝑥 𝑡 𝑦 ↔ 𝑦 𝑡 𝑥 ) → 𝑡 = { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑐 ∈ { 𝑞 ∈ ( Pairs ‘ 𝑉 ) ∣ ∀ 𝑎 ∈ 𝑉 ∀ 𝑏 ∈ 𝑉 ( 𝑞 = { 𝑎 , 𝑏 } → 𝑎 𝑡 𝑏 ) } 𝑐 = { 𝑥 , 𝑦 } } ) ) |
42 |
41
|
expcom |
⊢ ( 𝑡 ⊆ ( 𝑉 × 𝑉 ) → ( 𝑉 ∈ 𝑊 → ( ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( 𝑥 𝑡 𝑦 ↔ 𝑦 𝑡 𝑥 ) → 𝑡 = { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑐 ∈ { 𝑞 ∈ ( Pairs ‘ 𝑉 ) ∣ ∀ 𝑎 ∈ 𝑉 ∀ 𝑏 ∈ 𝑉 ( 𝑞 = { 𝑎 , 𝑏 } → 𝑎 𝑡 𝑏 ) } 𝑐 = { 𝑥 , 𝑦 } } ) ) ) |
43 |
42
|
com23 |
⊢ ( 𝑡 ⊆ ( 𝑉 × 𝑉 ) → ( ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( 𝑥 𝑡 𝑦 ↔ 𝑦 𝑡 𝑥 ) → ( 𝑉 ∈ 𝑊 → 𝑡 = { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑐 ∈ { 𝑞 ∈ ( Pairs ‘ 𝑉 ) ∣ ∀ 𝑎 ∈ 𝑉 ∀ 𝑏 ∈ 𝑉 ( 𝑞 = { 𝑎 , 𝑏 } → 𝑎 𝑡 𝑏 ) } 𝑐 = { 𝑥 , 𝑦 } } ) ) ) |
44 |
19 43
|
sylbi |
⊢ ( 𝑡 ∈ 𝒫 ( 𝑉 × 𝑉 ) → ( ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( 𝑥 𝑡 𝑦 ↔ 𝑦 𝑡 𝑥 ) → ( 𝑉 ∈ 𝑊 → 𝑡 = { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑐 ∈ { 𝑞 ∈ ( Pairs ‘ 𝑉 ) ∣ ∀ 𝑎 ∈ 𝑉 ∀ 𝑏 ∈ 𝑉 ( 𝑞 = { 𝑎 , 𝑏 } → 𝑎 𝑡 𝑏 ) } 𝑐 = { 𝑥 , 𝑦 } } ) ) ) |
45 |
44
|
imp31 |
⊢ ( ( ( 𝑡 ∈ 𝒫 ( 𝑉 × 𝑉 ) ∧ ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( 𝑥 𝑡 𝑦 ↔ 𝑦 𝑡 𝑥 ) ) ∧ 𝑉 ∈ 𝑊 ) → 𝑡 = { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑐 ∈ { 𝑞 ∈ ( Pairs ‘ 𝑉 ) ∣ ∀ 𝑎 ∈ 𝑉 ∀ 𝑏 ∈ 𝑉 ( 𝑞 = { 𝑎 , 𝑏 } → 𝑎 𝑡 𝑏 ) } 𝑐 = { 𝑥 , 𝑦 } } ) |
46 |
14 18 45
|
rspcedvd |
⊢ ( ( ( 𝑡 ∈ 𝒫 ( 𝑉 × 𝑉 ) ∧ ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( 𝑥 𝑡 𝑦 ↔ 𝑦 𝑡 𝑥 ) ) ∧ 𝑉 ∈ 𝑊 ) → ∃ 𝑓 ∈ 𝑃 𝑡 = { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑐 ∈ 𝑓 𝑐 = { 𝑥 , 𝑦 } } ) |
47 |
46
|
ex |
⊢ ( ( 𝑡 ∈ 𝒫 ( 𝑉 × 𝑉 ) ∧ ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( 𝑥 𝑡 𝑦 ↔ 𝑦 𝑡 𝑥 ) ) → ( 𝑉 ∈ 𝑊 → ∃ 𝑓 ∈ 𝑃 𝑡 = { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑐 ∈ 𝑓 𝑐 = { 𝑥 , 𝑦 } } ) ) |
48 |
10 47
|
sylbi |
⊢ ( 𝑡 ∈ 𝑅 → ( 𝑉 ∈ 𝑊 → ∃ 𝑓 ∈ 𝑃 𝑡 = { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑐 ∈ 𝑓 𝑐 = { 𝑥 , 𝑦 } } ) ) |
49 |
48
|
impcom |
⊢ ( ( 𝑉 ∈ 𝑊 ∧ 𝑡 ∈ 𝑅 ) → ∃ 𝑓 ∈ 𝑃 𝑡 = { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑐 ∈ 𝑓 𝑐 = { 𝑥 , 𝑦 } } ) |
50 |
1 2 3
|
sprsymrelfv |
⊢ ( 𝑓 ∈ 𝑃 → ( 𝐹 ‘ 𝑓 ) = { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑐 ∈ 𝑓 𝑐 = { 𝑥 , 𝑦 } } ) |
51 |
50
|
adantl |
⊢ ( ( ( 𝑉 ∈ 𝑊 ∧ 𝑡 ∈ 𝑅 ) ∧ 𝑓 ∈ 𝑃 ) → ( 𝐹 ‘ 𝑓 ) = { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑐 ∈ 𝑓 𝑐 = { 𝑥 , 𝑦 } } ) |
52 |
51
|
eqeq2d |
⊢ ( ( ( 𝑉 ∈ 𝑊 ∧ 𝑡 ∈ 𝑅 ) ∧ 𝑓 ∈ 𝑃 ) → ( 𝑡 = ( 𝐹 ‘ 𝑓 ) ↔ 𝑡 = { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑐 ∈ 𝑓 𝑐 = { 𝑥 , 𝑦 } } ) ) |
53 |
52
|
rexbidva |
⊢ ( ( 𝑉 ∈ 𝑊 ∧ 𝑡 ∈ 𝑅 ) → ( ∃ 𝑓 ∈ 𝑃 𝑡 = ( 𝐹 ‘ 𝑓 ) ↔ ∃ 𝑓 ∈ 𝑃 𝑡 = { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑐 ∈ 𝑓 𝑐 = { 𝑥 , 𝑦 } } ) ) |
54 |
49 53
|
mpbird |
⊢ ( ( 𝑉 ∈ 𝑊 ∧ 𝑡 ∈ 𝑅 ) → ∃ 𝑓 ∈ 𝑃 𝑡 = ( 𝐹 ‘ 𝑓 ) ) |
55 |
54
|
ralrimiva |
⊢ ( 𝑉 ∈ 𝑊 → ∀ 𝑡 ∈ 𝑅 ∃ 𝑓 ∈ 𝑃 𝑡 = ( 𝐹 ‘ 𝑓 ) ) |
56 |
|
dffo3 |
⊢ ( 𝐹 : 𝑃 –onto→ 𝑅 ↔ ( 𝐹 : 𝑃 ⟶ 𝑅 ∧ ∀ 𝑡 ∈ 𝑅 ∃ 𝑓 ∈ 𝑃 𝑡 = ( 𝐹 ‘ 𝑓 ) ) ) |
57 |
5 55 56
|
sylanbrc |
⊢ ( 𝑉 ∈ 𝑊 → 𝐹 : 𝑃 –onto→ 𝑅 ) |