Step |
Hyp |
Ref |
Expression |
1 |
|
funopab |
⊢ ( Fun { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } ) } ↔ ∀ 𝑥 ∃* 𝑦 ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } ) ) |
2 |
|
oveq1 |
⊢ ( 𝑖 = 𝑘 → ( 𝑖 ∈𝑔 𝑗 ) = ( 𝑘 ∈𝑔 𝑗 ) ) |
3 |
2
|
eqeq2d |
⊢ ( 𝑖 = 𝑘 → ( 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ↔ 𝑥 = ( 𝑘 ∈𝑔 𝑗 ) ) ) |
4 |
|
fveq2 |
⊢ ( 𝑖 = 𝑘 → ( 𝑓 ‘ 𝑖 ) = ( 𝑓 ‘ 𝑘 ) ) |
5 |
4
|
breq1d |
⊢ ( 𝑖 = 𝑘 → ( ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) ↔ ( 𝑓 ‘ 𝑘 ) 𝐸 ( 𝑓 ‘ 𝑗 ) ) ) |
6 |
5
|
rabbidv |
⊢ ( 𝑖 = 𝑘 → { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑘 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } ) |
7 |
6
|
eqeq2d |
⊢ ( 𝑖 = 𝑘 → ( 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } ↔ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑘 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } ) ) |
8 |
3 7
|
anbi12d |
⊢ ( 𝑖 = 𝑘 → ( ( 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } ) ↔ ( 𝑥 = ( 𝑘 ∈𝑔 𝑗 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑘 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } ) ) ) |
9 |
|
oveq2 |
⊢ ( 𝑗 = 𝑙 → ( 𝑘 ∈𝑔 𝑗 ) = ( 𝑘 ∈𝑔 𝑙 ) ) |
10 |
9
|
eqeq2d |
⊢ ( 𝑗 = 𝑙 → ( 𝑥 = ( 𝑘 ∈𝑔 𝑗 ) ↔ 𝑥 = ( 𝑘 ∈𝑔 𝑙 ) ) ) |
11 |
|
fveq2 |
⊢ ( 𝑗 = 𝑙 → ( 𝑓 ‘ 𝑗 ) = ( 𝑓 ‘ 𝑙 ) ) |
12 |
11
|
breq2d |
⊢ ( 𝑗 = 𝑙 → ( ( 𝑓 ‘ 𝑘 ) 𝐸 ( 𝑓 ‘ 𝑗 ) ↔ ( 𝑓 ‘ 𝑘 ) 𝐸 ( 𝑓 ‘ 𝑙 ) ) ) |
13 |
12
|
rabbidv |
⊢ ( 𝑗 = 𝑙 → { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑘 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑘 ) 𝐸 ( 𝑓 ‘ 𝑙 ) } ) |
14 |
13
|
eqeq2d |
⊢ ( 𝑗 = 𝑙 → ( 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑘 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } ↔ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑘 ) 𝐸 ( 𝑓 ‘ 𝑙 ) } ) ) |
15 |
10 14
|
anbi12d |
⊢ ( 𝑗 = 𝑙 → ( ( 𝑥 = ( 𝑘 ∈𝑔 𝑗 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑘 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } ) ↔ ( 𝑥 = ( 𝑘 ∈𝑔 𝑙 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑘 ) 𝐸 ( 𝑓 ‘ 𝑙 ) } ) ) ) |
16 |
8 15
|
cbvrex2vw |
⊢ ( ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } ) ↔ ∃ 𝑘 ∈ ω ∃ 𝑙 ∈ ω ( 𝑥 = ( 𝑘 ∈𝑔 𝑙 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑘 ) 𝐸 ( 𝑓 ‘ 𝑙 ) } ) ) |
17 |
|
eqtr2 |
⊢ ( ( 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ∧ 𝑥 = ( 𝑘 ∈𝑔 𝑙 ) ) → ( 𝑖 ∈𝑔 𝑗 ) = ( 𝑘 ∈𝑔 𝑙 ) ) |
18 |
|
goeleq12bg |
⊢ ( ( ( 𝑘 ∈ ω ∧ 𝑙 ∈ ω ) ∧ ( 𝑖 ∈ ω ∧ 𝑗 ∈ ω ) ) → ( ( 𝑖 ∈𝑔 𝑗 ) = ( 𝑘 ∈𝑔 𝑙 ) ↔ ( 𝑖 = 𝑘 ∧ 𝑗 = 𝑙 ) ) ) |
19 |
4
|
adantr |
⊢ ( ( 𝑖 = 𝑘 ∧ 𝑗 = 𝑙 ) → ( 𝑓 ‘ 𝑖 ) = ( 𝑓 ‘ 𝑘 ) ) |
20 |
19
|
eqcomd |
⊢ ( ( 𝑖 = 𝑘 ∧ 𝑗 = 𝑙 ) → ( 𝑓 ‘ 𝑘 ) = ( 𝑓 ‘ 𝑖 ) ) |
21 |
11
|
adantl |
⊢ ( ( 𝑖 = 𝑘 ∧ 𝑗 = 𝑙 ) → ( 𝑓 ‘ 𝑗 ) = ( 𝑓 ‘ 𝑙 ) ) |
22 |
21
|
eqcomd |
⊢ ( ( 𝑖 = 𝑘 ∧ 𝑗 = 𝑙 ) → ( 𝑓 ‘ 𝑙 ) = ( 𝑓 ‘ 𝑗 ) ) |
23 |
20 22
|
breq12d |
⊢ ( ( 𝑖 = 𝑘 ∧ 𝑗 = 𝑙 ) → ( ( 𝑓 ‘ 𝑘 ) 𝐸 ( 𝑓 ‘ 𝑙 ) ↔ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) ) ) |
24 |
23
|
rabbidv |
⊢ ( ( 𝑖 = 𝑘 ∧ 𝑗 = 𝑙 ) → { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑘 ) 𝐸 ( 𝑓 ‘ 𝑙 ) } = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } ) |
25 |
|
eqeq12 |
⊢ ( ( 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑘 ) 𝐸 ( 𝑓 ‘ 𝑙 ) } ∧ 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } ) → ( 𝑦 = 𝑧 ↔ { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑘 ) 𝐸 ( 𝑓 ‘ 𝑙 ) } = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } ) ) |
26 |
24 25
|
syl5ibrcom |
⊢ ( ( 𝑖 = 𝑘 ∧ 𝑗 = 𝑙 ) → ( ( 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑘 ) 𝐸 ( 𝑓 ‘ 𝑙 ) } ∧ 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } ) → 𝑦 = 𝑧 ) ) |
27 |
26
|
expd |
⊢ ( ( 𝑖 = 𝑘 ∧ 𝑗 = 𝑙 ) → ( 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑘 ) 𝐸 ( 𝑓 ‘ 𝑙 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } → 𝑦 = 𝑧 ) ) ) |
28 |
18 27
|
syl6bi |
⊢ ( ( ( 𝑘 ∈ ω ∧ 𝑙 ∈ ω ) ∧ ( 𝑖 ∈ ω ∧ 𝑗 ∈ ω ) ) → ( ( 𝑖 ∈𝑔 𝑗 ) = ( 𝑘 ∈𝑔 𝑙 ) → ( 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑘 ) 𝐸 ( 𝑓 ‘ 𝑙 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } → 𝑦 = 𝑧 ) ) ) ) |
29 |
17 28
|
syl5 |
⊢ ( ( ( 𝑘 ∈ ω ∧ 𝑙 ∈ ω ) ∧ ( 𝑖 ∈ ω ∧ 𝑗 ∈ ω ) ) → ( ( 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ∧ 𝑥 = ( 𝑘 ∈𝑔 𝑙 ) ) → ( 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑘 ) 𝐸 ( 𝑓 ‘ 𝑙 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } → 𝑦 = 𝑧 ) ) ) ) |
30 |
29
|
expd |
⊢ ( ( ( 𝑘 ∈ ω ∧ 𝑙 ∈ ω ) ∧ ( 𝑖 ∈ ω ∧ 𝑗 ∈ ω ) ) → ( 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) → ( 𝑥 = ( 𝑘 ∈𝑔 𝑙 ) → ( 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑘 ) 𝐸 ( 𝑓 ‘ 𝑙 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } → 𝑦 = 𝑧 ) ) ) ) ) |
31 |
30
|
imp4a |
⊢ ( ( ( 𝑘 ∈ ω ∧ 𝑙 ∈ ω ) ∧ ( 𝑖 ∈ ω ∧ 𝑗 ∈ ω ) ) → ( 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) → ( ( 𝑥 = ( 𝑘 ∈𝑔 𝑙 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑘 ) 𝐸 ( 𝑓 ‘ 𝑙 ) } ) → ( 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } → 𝑦 = 𝑧 ) ) ) ) |
32 |
31
|
com34 |
⊢ ( ( ( 𝑘 ∈ ω ∧ 𝑙 ∈ ω ) ∧ ( 𝑖 ∈ ω ∧ 𝑗 ∈ ω ) ) → ( 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) → ( 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } → ( ( 𝑥 = ( 𝑘 ∈𝑔 𝑙 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑘 ) 𝐸 ( 𝑓 ‘ 𝑙 ) } ) → 𝑦 = 𝑧 ) ) ) ) |
33 |
32
|
impd |
⊢ ( ( ( 𝑘 ∈ ω ∧ 𝑙 ∈ ω ) ∧ ( 𝑖 ∈ ω ∧ 𝑗 ∈ ω ) ) → ( ( 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } ) → ( ( 𝑥 = ( 𝑘 ∈𝑔 𝑙 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑘 ) 𝐸 ( 𝑓 ‘ 𝑙 ) } ) → 𝑦 = 𝑧 ) ) ) |
34 |
33
|
rexlimdvva |
⊢ ( ( 𝑘 ∈ ω ∧ 𝑙 ∈ ω ) → ( ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } ) → ( ( 𝑥 = ( 𝑘 ∈𝑔 𝑙 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑘 ) 𝐸 ( 𝑓 ‘ 𝑙 ) } ) → 𝑦 = 𝑧 ) ) ) |
35 |
34
|
com23 |
⊢ ( ( 𝑘 ∈ ω ∧ 𝑙 ∈ ω ) → ( ( 𝑥 = ( 𝑘 ∈𝑔 𝑙 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑘 ) 𝐸 ( 𝑓 ‘ 𝑙 ) } ) → ( ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } ) → 𝑦 = 𝑧 ) ) ) |
36 |
35
|
rexlimivv |
⊢ ( ∃ 𝑘 ∈ ω ∃ 𝑙 ∈ ω ( 𝑥 = ( 𝑘 ∈𝑔 𝑙 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑘 ) 𝐸 ( 𝑓 ‘ 𝑙 ) } ) → ( ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } ) → 𝑦 = 𝑧 ) ) |
37 |
16 36
|
sylbi |
⊢ ( ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } ) → ( ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } ) → 𝑦 = 𝑧 ) ) |
38 |
37
|
imp |
⊢ ( ( ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } ) ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } ) ) → 𝑦 = 𝑧 ) |
39 |
38
|
gen2 |
⊢ ∀ 𝑦 ∀ 𝑧 ( ( ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } ) ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } ) ) → 𝑦 = 𝑧 ) |
40 |
|
eqeq1 |
⊢ ( 𝑦 = 𝑧 → ( 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } ↔ 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } ) ) |
41 |
40
|
anbi2d |
⊢ ( 𝑦 = 𝑧 → ( ( 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } ) ↔ ( 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } ) ) ) |
42 |
41
|
2rexbidv |
⊢ ( 𝑦 = 𝑧 → ( ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } ) ↔ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } ) ) ) |
43 |
42
|
mo4 |
⊢ ( ∃* 𝑦 ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } ) ↔ ∀ 𝑦 ∀ 𝑧 ( ( ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } ) ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } ) ) → 𝑦 = 𝑧 ) ) |
44 |
39 43
|
mpbir |
⊢ ∃* 𝑦 ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } ) |
45 |
1 44
|
mpgbir |
⊢ Fun { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } ) } |
46 |
|
eqid |
⊢ ( 𝑀 Sat 𝐸 ) = ( 𝑀 Sat 𝐸 ) |
47 |
46
|
satfv0 |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → ( ( 𝑀 Sat 𝐸 ) ‘ ∅ ) = { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } ) } ) |
48 |
47
|
funeqd |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ ∅ ) ↔ Fun { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖 ∈𝑔 𝑗 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑓 ‘ 𝑖 ) 𝐸 ( 𝑓 ‘ 𝑗 ) } ) } ) ) |
49 |
45 48
|
mpbiri |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → Fun ( ( 𝑀 Sat 𝐸 ) ‘ ∅ ) ) |