Step |
Hyp |
Ref |
Expression |
1 |
|
rexcom4 |
⊢ ( ∃ 𝑣 ∈ 𝑉 ∃ 𝑦 ( 𝑧 = 𝐴 ∧ 𝑦 = 𝐵 ) ↔ ∃ 𝑦 ∃ 𝑣 ∈ 𝑉 ( 𝑧 = 𝐴 ∧ 𝑦 = 𝐵 ) ) |
2 |
|
rexcom4 |
⊢ ( ∃ 𝑖 ∈ 𝐼 ∃ 𝑦 ( 𝑧 = 𝐶 ∧ 𝑦 = 𝐷 ) ↔ ∃ 𝑦 ∃ 𝑖 ∈ 𝐼 ( 𝑧 = 𝐶 ∧ 𝑦 = 𝐷 ) ) |
3 |
1 2
|
orbi12i |
⊢ ( ( ∃ 𝑣 ∈ 𝑉 ∃ 𝑦 ( 𝑧 = 𝐴 ∧ 𝑦 = 𝐵 ) ∨ ∃ 𝑖 ∈ 𝐼 ∃ 𝑦 ( 𝑧 = 𝐶 ∧ 𝑦 = 𝐷 ) ) ↔ ( ∃ 𝑦 ∃ 𝑣 ∈ 𝑉 ( 𝑧 = 𝐴 ∧ 𝑦 = 𝐵 ) ∨ ∃ 𝑦 ∃ 𝑖 ∈ 𝐼 ( 𝑧 = 𝐶 ∧ 𝑦 = 𝐷 ) ) ) |
4 |
|
19.43 |
⊢ ( ∃ 𝑦 ( ∃ 𝑣 ∈ 𝑉 ( 𝑧 = 𝐴 ∧ 𝑦 = 𝐵 ) ∨ ∃ 𝑖 ∈ 𝐼 ( 𝑧 = 𝐶 ∧ 𝑦 = 𝐷 ) ) ↔ ( ∃ 𝑦 ∃ 𝑣 ∈ 𝑉 ( 𝑧 = 𝐴 ∧ 𝑦 = 𝐵 ) ∨ ∃ 𝑦 ∃ 𝑖 ∈ 𝐼 ( 𝑧 = 𝐶 ∧ 𝑦 = 𝐷 ) ) ) |
5 |
3 4
|
bitr4i |
⊢ ( ( ∃ 𝑣 ∈ 𝑉 ∃ 𝑦 ( 𝑧 = 𝐴 ∧ 𝑦 = 𝐵 ) ∨ ∃ 𝑖 ∈ 𝐼 ∃ 𝑦 ( 𝑧 = 𝐶 ∧ 𝑦 = 𝐷 ) ) ↔ ∃ 𝑦 ( ∃ 𝑣 ∈ 𝑉 ( 𝑧 = 𝐴 ∧ 𝑦 = 𝐵 ) ∨ ∃ 𝑖 ∈ 𝐼 ( 𝑧 = 𝐶 ∧ 𝑦 = 𝐷 ) ) ) |
6 |
5
|
rexbii |
⊢ ( ∃ 𝑢 ∈ 𝑈 ( ∃ 𝑣 ∈ 𝑉 ∃ 𝑦 ( 𝑧 = 𝐴 ∧ 𝑦 = 𝐵 ) ∨ ∃ 𝑖 ∈ 𝐼 ∃ 𝑦 ( 𝑧 = 𝐶 ∧ 𝑦 = 𝐷 ) ) ↔ ∃ 𝑢 ∈ 𝑈 ∃ 𝑦 ( ∃ 𝑣 ∈ 𝑉 ( 𝑧 = 𝐴 ∧ 𝑦 = 𝐵 ) ∨ ∃ 𝑖 ∈ 𝐼 ( 𝑧 = 𝐶 ∧ 𝑦 = 𝐷 ) ) ) |
7 |
|
rexcom4 |
⊢ ( ∃ 𝑢 ∈ 𝑈 ∃ 𝑦 ( ∃ 𝑣 ∈ 𝑉 ( 𝑧 = 𝐴 ∧ 𝑦 = 𝐵 ) ∨ ∃ 𝑖 ∈ 𝐼 ( 𝑧 = 𝐶 ∧ 𝑦 = 𝐷 ) ) ↔ ∃ 𝑦 ∃ 𝑢 ∈ 𝑈 ( ∃ 𝑣 ∈ 𝑉 ( 𝑧 = 𝐴 ∧ 𝑦 = 𝐵 ) ∨ ∃ 𝑖 ∈ 𝐼 ( 𝑧 = 𝐶 ∧ 𝑦 = 𝐷 ) ) ) |
8 |
6 7
|
bitri |
⊢ ( ∃ 𝑢 ∈ 𝑈 ( ∃ 𝑣 ∈ 𝑉 ∃ 𝑦 ( 𝑧 = 𝐴 ∧ 𝑦 = 𝐵 ) ∨ ∃ 𝑖 ∈ 𝐼 ∃ 𝑦 ( 𝑧 = 𝐶 ∧ 𝑦 = 𝐷 ) ) ↔ ∃ 𝑦 ∃ 𝑢 ∈ 𝑈 ( ∃ 𝑣 ∈ 𝑉 ( 𝑧 = 𝐴 ∧ 𝑦 = 𝐵 ) ∨ ∃ 𝑖 ∈ 𝐼 ( 𝑧 = 𝐶 ∧ 𝑦 = 𝐷 ) ) ) |
9 |
|
simpl |
⊢ ( ( 𝑧 = 𝐴 ∧ 𝑦 = 𝐵 ) → 𝑧 = 𝐴 ) |
10 |
9
|
exlimiv |
⊢ ( ∃ 𝑦 ( 𝑧 = 𝐴 ∧ 𝑦 = 𝐵 ) → 𝑧 = 𝐴 ) |
11 |
|
elisset |
⊢ ( 𝐵 ∈ 𝑋 → ∃ 𝑦 𝑦 = 𝐵 ) |
12 |
|
ibar |
⊢ ( 𝑧 = 𝐴 → ( 𝑦 = 𝐵 ↔ ( 𝑧 = 𝐴 ∧ 𝑦 = 𝐵 ) ) ) |
13 |
12
|
bicomd |
⊢ ( 𝑧 = 𝐴 → ( ( 𝑧 = 𝐴 ∧ 𝑦 = 𝐵 ) ↔ 𝑦 = 𝐵 ) ) |
14 |
13
|
exbidv |
⊢ ( 𝑧 = 𝐴 → ( ∃ 𝑦 ( 𝑧 = 𝐴 ∧ 𝑦 = 𝐵 ) ↔ ∃ 𝑦 𝑦 = 𝐵 ) ) |
15 |
11 14
|
syl5ibrcom |
⊢ ( 𝐵 ∈ 𝑋 → ( 𝑧 = 𝐴 → ∃ 𝑦 ( 𝑧 = 𝐴 ∧ 𝑦 = 𝐵 ) ) ) |
16 |
10 15
|
impbid2 |
⊢ ( 𝐵 ∈ 𝑋 → ( ∃ 𝑦 ( 𝑧 = 𝐴 ∧ 𝑦 = 𝐵 ) ↔ 𝑧 = 𝐴 ) ) |
17 |
16
|
ralrexbid |
⊢ ( ∀ 𝑣 ∈ 𝑉 𝐵 ∈ 𝑋 → ( ∃ 𝑣 ∈ 𝑉 ∃ 𝑦 ( 𝑧 = 𝐴 ∧ 𝑦 = 𝐵 ) ↔ ∃ 𝑣 ∈ 𝑉 𝑧 = 𝐴 ) ) |
18 |
17
|
adantr |
⊢ ( ( ∀ 𝑣 ∈ 𝑉 𝐵 ∈ 𝑋 ∧ ∀ 𝑖 ∈ 𝐼 𝐷 ∈ 𝑊 ) → ( ∃ 𝑣 ∈ 𝑉 ∃ 𝑦 ( 𝑧 = 𝐴 ∧ 𝑦 = 𝐵 ) ↔ ∃ 𝑣 ∈ 𝑉 𝑧 = 𝐴 ) ) |
19 |
|
simpl |
⊢ ( ( 𝑧 = 𝐶 ∧ 𝑦 = 𝐷 ) → 𝑧 = 𝐶 ) |
20 |
19
|
exlimiv |
⊢ ( ∃ 𝑦 ( 𝑧 = 𝐶 ∧ 𝑦 = 𝐷 ) → 𝑧 = 𝐶 ) |
21 |
|
elisset |
⊢ ( 𝐷 ∈ 𝑊 → ∃ 𝑦 𝑦 = 𝐷 ) |
22 |
|
ibar |
⊢ ( 𝑧 = 𝐶 → ( 𝑦 = 𝐷 ↔ ( 𝑧 = 𝐶 ∧ 𝑦 = 𝐷 ) ) ) |
23 |
22
|
bicomd |
⊢ ( 𝑧 = 𝐶 → ( ( 𝑧 = 𝐶 ∧ 𝑦 = 𝐷 ) ↔ 𝑦 = 𝐷 ) ) |
24 |
23
|
exbidv |
⊢ ( 𝑧 = 𝐶 → ( ∃ 𝑦 ( 𝑧 = 𝐶 ∧ 𝑦 = 𝐷 ) ↔ ∃ 𝑦 𝑦 = 𝐷 ) ) |
25 |
21 24
|
syl5ibrcom |
⊢ ( 𝐷 ∈ 𝑊 → ( 𝑧 = 𝐶 → ∃ 𝑦 ( 𝑧 = 𝐶 ∧ 𝑦 = 𝐷 ) ) ) |
26 |
20 25
|
impbid2 |
⊢ ( 𝐷 ∈ 𝑊 → ( ∃ 𝑦 ( 𝑧 = 𝐶 ∧ 𝑦 = 𝐷 ) ↔ 𝑧 = 𝐶 ) ) |
27 |
26
|
ralrexbid |
⊢ ( ∀ 𝑖 ∈ 𝐼 𝐷 ∈ 𝑊 → ( ∃ 𝑖 ∈ 𝐼 ∃ 𝑦 ( 𝑧 = 𝐶 ∧ 𝑦 = 𝐷 ) ↔ ∃ 𝑖 ∈ 𝐼 𝑧 = 𝐶 ) ) |
28 |
27
|
adantl |
⊢ ( ( ∀ 𝑣 ∈ 𝑉 𝐵 ∈ 𝑋 ∧ ∀ 𝑖 ∈ 𝐼 𝐷 ∈ 𝑊 ) → ( ∃ 𝑖 ∈ 𝐼 ∃ 𝑦 ( 𝑧 = 𝐶 ∧ 𝑦 = 𝐷 ) ↔ ∃ 𝑖 ∈ 𝐼 𝑧 = 𝐶 ) ) |
29 |
18 28
|
orbi12d |
⊢ ( ( ∀ 𝑣 ∈ 𝑉 𝐵 ∈ 𝑋 ∧ ∀ 𝑖 ∈ 𝐼 𝐷 ∈ 𝑊 ) → ( ( ∃ 𝑣 ∈ 𝑉 ∃ 𝑦 ( 𝑧 = 𝐴 ∧ 𝑦 = 𝐵 ) ∨ ∃ 𝑖 ∈ 𝐼 ∃ 𝑦 ( 𝑧 = 𝐶 ∧ 𝑦 = 𝐷 ) ) ↔ ( ∃ 𝑣 ∈ 𝑉 𝑧 = 𝐴 ∨ ∃ 𝑖 ∈ 𝐼 𝑧 = 𝐶 ) ) ) |
30 |
29
|
ralrexbid |
⊢ ( ∀ 𝑢 ∈ 𝑈 ( ∀ 𝑣 ∈ 𝑉 𝐵 ∈ 𝑋 ∧ ∀ 𝑖 ∈ 𝐼 𝐷 ∈ 𝑊 ) → ( ∃ 𝑢 ∈ 𝑈 ( ∃ 𝑣 ∈ 𝑉 ∃ 𝑦 ( 𝑧 = 𝐴 ∧ 𝑦 = 𝐵 ) ∨ ∃ 𝑖 ∈ 𝐼 ∃ 𝑦 ( 𝑧 = 𝐶 ∧ 𝑦 = 𝐷 ) ) ↔ ∃ 𝑢 ∈ 𝑈 ( ∃ 𝑣 ∈ 𝑉 𝑧 = 𝐴 ∨ ∃ 𝑖 ∈ 𝐼 𝑧 = 𝐶 ) ) ) |
31 |
8 30
|
bitr3id |
⊢ ( ∀ 𝑢 ∈ 𝑈 ( ∀ 𝑣 ∈ 𝑉 𝐵 ∈ 𝑋 ∧ ∀ 𝑖 ∈ 𝐼 𝐷 ∈ 𝑊 ) → ( ∃ 𝑦 ∃ 𝑢 ∈ 𝑈 ( ∃ 𝑣 ∈ 𝑉 ( 𝑧 = 𝐴 ∧ 𝑦 = 𝐵 ) ∨ ∃ 𝑖 ∈ 𝐼 ( 𝑧 = 𝐶 ∧ 𝑦 = 𝐷 ) ) ↔ ∃ 𝑢 ∈ 𝑈 ( ∃ 𝑣 ∈ 𝑉 𝑧 = 𝐴 ∨ ∃ 𝑖 ∈ 𝐼 𝑧 = 𝐶 ) ) ) |
32 |
|
eqeq1 |
⊢ ( 𝑥 = 𝑧 → ( 𝑥 = 𝐴 ↔ 𝑧 = 𝐴 ) ) |
33 |
32
|
anbi1d |
⊢ ( 𝑥 = 𝑧 → ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ↔ ( 𝑧 = 𝐴 ∧ 𝑦 = 𝐵 ) ) ) |
34 |
33
|
rexbidv |
⊢ ( 𝑥 = 𝑧 → ( ∃ 𝑣 ∈ 𝑉 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ↔ ∃ 𝑣 ∈ 𝑉 ( 𝑧 = 𝐴 ∧ 𝑦 = 𝐵 ) ) ) |
35 |
|
eqeq1 |
⊢ ( 𝑥 = 𝑧 → ( 𝑥 = 𝐶 ↔ 𝑧 = 𝐶 ) ) |
36 |
35
|
anbi1d |
⊢ ( 𝑥 = 𝑧 → ( ( 𝑥 = 𝐶 ∧ 𝑦 = 𝐷 ) ↔ ( 𝑧 = 𝐶 ∧ 𝑦 = 𝐷 ) ) ) |
37 |
36
|
rexbidv |
⊢ ( 𝑥 = 𝑧 → ( ∃ 𝑖 ∈ 𝐼 ( 𝑥 = 𝐶 ∧ 𝑦 = 𝐷 ) ↔ ∃ 𝑖 ∈ 𝐼 ( 𝑧 = 𝐶 ∧ 𝑦 = 𝐷 ) ) ) |
38 |
34 37
|
orbi12d |
⊢ ( 𝑥 = 𝑧 → ( ( ∃ 𝑣 ∈ 𝑉 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∨ ∃ 𝑖 ∈ 𝐼 ( 𝑥 = 𝐶 ∧ 𝑦 = 𝐷 ) ) ↔ ( ∃ 𝑣 ∈ 𝑉 ( 𝑧 = 𝐴 ∧ 𝑦 = 𝐵 ) ∨ ∃ 𝑖 ∈ 𝐼 ( 𝑧 = 𝐶 ∧ 𝑦 = 𝐷 ) ) ) ) |
39 |
38
|
rexbidv |
⊢ ( 𝑥 = 𝑧 → ( ∃ 𝑢 ∈ 𝑈 ( ∃ 𝑣 ∈ 𝑉 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∨ ∃ 𝑖 ∈ 𝐼 ( 𝑥 = 𝐶 ∧ 𝑦 = 𝐷 ) ) ↔ ∃ 𝑢 ∈ 𝑈 ( ∃ 𝑣 ∈ 𝑉 ( 𝑧 = 𝐴 ∧ 𝑦 = 𝐵 ) ∨ ∃ 𝑖 ∈ 𝐼 ( 𝑧 = 𝐶 ∧ 𝑦 = 𝐷 ) ) ) ) |
40 |
39
|
dmopabelb |
⊢ ( 𝑧 ∈ V → ( 𝑧 ∈ dom { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ∈ 𝑈 ( ∃ 𝑣 ∈ 𝑉 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∨ ∃ 𝑖 ∈ 𝐼 ( 𝑥 = 𝐶 ∧ 𝑦 = 𝐷 ) ) } ↔ ∃ 𝑦 ∃ 𝑢 ∈ 𝑈 ( ∃ 𝑣 ∈ 𝑉 ( 𝑧 = 𝐴 ∧ 𝑦 = 𝐵 ) ∨ ∃ 𝑖 ∈ 𝐼 ( 𝑧 = 𝐶 ∧ 𝑦 = 𝐷 ) ) ) ) |
41 |
40
|
elv |
⊢ ( 𝑧 ∈ dom { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ∈ 𝑈 ( ∃ 𝑣 ∈ 𝑉 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∨ ∃ 𝑖 ∈ 𝐼 ( 𝑥 = 𝐶 ∧ 𝑦 = 𝐷 ) ) } ↔ ∃ 𝑦 ∃ 𝑢 ∈ 𝑈 ( ∃ 𝑣 ∈ 𝑉 ( 𝑧 = 𝐴 ∧ 𝑦 = 𝐵 ) ∨ ∃ 𝑖 ∈ 𝐼 ( 𝑧 = 𝐶 ∧ 𝑦 = 𝐷 ) ) ) |
42 |
|
vex |
⊢ 𝑧 ∈ V |
43 |
32
|
rexbidv |
⊢ ( 𝑥 = 𝑧 → ( ∃ 𝑣 ∈ 𝑉 𝑥 = 𝐴 ↔ ∃ 𝑣 ∈ 𝑉 𝑧 = 𝐴 ) ) |
44 |
35
|
rexbidv |
⊢ ( 𝑥 = 𝑧 → ( ∃ 𝑖 ∈ 𝐼 𝑥 = 𝐶 ↔ ∃ 𝑖 ∈ 𝐼 𝑧 = 𝐶 ) ) |
45 |
43 44
|
orbi12d |
⊢ ( 𝑥 = 𝑧 → ( ( ∃ 𝑣 ∈ 𝑉 𝑥 = 𝐴 ∨ ∃ 𝑖 ∈ 𝐼 𝑥 = 𝐶 ) ↔ ( ∃ 𝑣 ∈ 𝑉 𝑧 = 𝐴 ∨ ∃ 𝑖 ∈ 𝐼 𝑧 = 𝐶 ) ) ) |
46 |
45
|
rexbidv |
⊢ ( 𝑥 = 𝑧 → ( ∃ 𝑢 ∈ 𝑈 ( ∃ 𝑣 ∈ 𝑉 𝑥 = 𝐴 ∨ ∃ 𝑖 ∈ 𝐼 𝑥 = 𝐶 ) ↔ ∃ 𝑢 ∈ 𝑈 ( ∃ 𝑣 ∈ 𝑉 𝑧 = 𝐴 ∨ ∃ 𝑖 ∈ 𝐼 𝑧 = 𝐶 ) ) ) |
47 |
42 46
|
elab |
⊢ ( 𝑧 ∈ { 𝑥 ∣ ∃ 𝑢 ∈ 𝑈 ( ∃ 𝑣 ∈ 𝑉 𝑥 = 𝐴 ∨ ∃ 𝑖 ∈ 𝐼 𝑥 = 𝐶 ) } ↔ ∃ 𝑢 ∈ 𝑈 ( ∃ 𝑣 ∈ 𝑉 𝑧 = 𝐴 ∨ ∃ 𝑖 ∈ 𝐼 𝑧 = 𝐶 ) ) |
48 |
31 41 47
|
3bitr4g |
⊢ ( ∀ 𝑢 ∈ 𝑈 ( ∀ 𝑣 ∈ 𝑉 𝐵 ∈ 𝑋 ∧ ∀ 𝑖 ∈ 𝐼 𝐷 ∈ 𝑊 ) → ( 𝑧 ∈ dom { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ∈ 𝑈 ( ∃ 𝑣 ∈ 𝑉 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∨ ∃ 𝑖 ∈ 𝐼 ( 𝑥 = 𝐶 ∧ 𝑦 = 𝐷 ) ) } ↔ 𝑧 ∈ { 𝑥 ∣ ∃ 𝑢 ∈ 𝑈 ( ∃ 𝑣 ∈ 𝑉 𝑥 = 𝐴 ∨ ∃ 𝑖 ∈ 𝐼 𝑥 = 𝐶 ) } ) ) |
49 |
48
|
eqrdv |
⊢ ( ∀ 𝑢 ∈ 𝑈 ( ∀ 𝑣 ∈ 𝑉 𝐵 ∈ 𝑋 ∧ ∀ 𝑖 ∈ 𝐼 𝐷 ∈ 𝑊 ) → dom { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ∈ 𝑈 ( ∃ 𝑣 ∈ 𝑉 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∨ ∃ 𝑖 ∈ 𝐼 ( 𝑥 = 𝐶 ∧ 𝑦 = 𝐷 ) ) } = { 𝑥 ∣ ∃ 𝑢 ∈ 𝑈 ( ∃ 𝑣 ∈ 𝑉 𝑥 = 𝐴 ∨ ∃ 𝑖 ∈ 𝐼 𝑥 = 𝐶 ) } ) |