Step |
Hyp |
Ref |
Expression |
1 |
|
sategoelfvb.s |
⊢ 𝐸 = ( 𝑀 Sat∈ ( 𝐴 ∈𝑔 𝐵 ) ) |
2 |
|
ovexd |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 ∈𝑔 𝐵 ) ∈ V ) |
3 |
|
simpl |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → 𝐴 ∈ ω ) |
4 |
|
opeq1 |
⊢ ( 𝑎 = 𝐴 → 〈 𝑎 , 𝑏 〉 = 〈 𝐴 , 𝑏 〉 ) |
5 |
4
|
opeq2d |
⊢ ( 𝑎 = 𝐴 → 〈 ∅ , 〈 𝑎 , 𝑏 〉 〉 = 〈 ∅ , 〈 𝐴 , 𝑏 〉 〉 ) |
6 |
5
|
eqeq2d |
⊢ ( 𝑎 = 𝐴 → ( 〈 ∅ , 〈 𝐴 , 𝐵 〉 〉 = 〈 ∅ , 〈 𝑎 , 𝑏 〉 〉 ↔ 〈 ∅ , 〈 𝐴 , 𝐵 〉 〉 = 〈 ∅ , 〈 𝐴 , 𝑏 〉 〉 ) ) |
7 |
6
|
rexbidv |
⊢ ( 𝑎 = 𝐴 → ( ∃ 𝑏 ∈ ω 〈 ∅ , 〈 𝐴 , 𝐵 〉 〉 = 〈 ∅ , 〈 𝑎 , 𝑏 〉 〉 ↔ ∃ 𝑏 ∈ ω 〈 ∅ , 〈 𝐴 , 𝐵 〉 〉 = 〈 ∅ , 〈 𝐴 , 𝑏 〉 〉 ) ) |
8 |
7
|
adantl |
⊢ ( ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ∧ 𝑎 = 𝐴 ) → ( ∃ 𝑏 ∈ ω 〈 ∅ , 〈 𝐴 , 𝐵 〉 〉 = 〈 ∅ , 〈 𝑎 , 𝑏 〉 〉 ↔ ∃ 𝑏 ∈ ω 〈 ∅ , 〈 𝐴 , 𝐵 〉 〉 = 〈 ∅ , 〈 𝐴 , 𝑏 〉 〉 ) ) |
9 |
|
simpr |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → 𝐵 ∈ ω ) |
10 |
|
opeq2 |
⊢ ( 𝑏 = 𝐵 → 〈 𝐴 , 𝑏 〉 = 〈 𝐴 , 𝐵 〉 ) |
11 |
10
|
opeq2d |
⊢ ( 𝑏 = 𝐵 → 〈 ∅ , 〈 𝐴 , 𝑏 〉 〉 = 〈 ∅ , 〈 𝐴 , 𝐵 〉 〉 ) |
12 |
11
|
eqeq2d |
⊢ ( 𝑏 = 𝐵 → ( 〈 ∅ , 〈 𝐴 , 𝐵 〉 〉 = 〈 ∅ , 〈 𝐴 , 𝑏 〉 〉 ↔ 〈 ∅ , 〈 𝐴 , 𝐵 〉 〉 = 〈 ∅ , 〈 𝐴 , 𝐵 〉 〉 ) ) |
13 |
12
|
adantl |
⊢ ( ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ∧ 𝑏 = 𝐵 ) → ( 〈 ∅ , 〈 𝐴 , 𝐵 〉 〉 = 〈 ∅ , 〈 𝐴 , 𝑏 〉 〉 ↔ 〈 ∅ , 〈 𝐴 , 𝐵 〉 〉 = 〈 ∅ , 〈 𝐴 , 𝐵 〉 〉 ) ) |
14 |
|
eqidd |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → 〈 ∅ , 〈 𝐴 , 𝐵 〉 〉 = 〈 ∅ , 〈 𝐴 , 𝐵 〉 〉 ) |
15 |
9 13 14
|
rspcedvd |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ∃ 𝑏 ∈ ω 〈 ∅ , 〈 𝐴 , 𝐵 〉 〉 = 〈 ∅ , 〈 𝐴 , 𝑏 〉 〉 ) |
16 |
3 8 15
|
rspcedvd |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ∃ 𝑎 ∈ ω ∃ 𝑏 ∈ ω 〈 ∅ , 〈 𝐴 , 𝐵 〉 〉 = 〈 ∅ , 〈 𝑎 , 𝑏 〉 〉 ) |
17 |
|
goel |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 ∈𝑔 𝐵 ) = 〈 ∅ , 〈 𝐴 , 𝐵 〉 〉 ) |
18 |
|
goel |
⊢ ( ( 𝑎 ∈ ω ∧ 𝑏 ∈ ω ) → ( 𝑎 ∈𝑔 𝑏 ) = 〈 ∅ , 〈 𝑎 , 𝑏 〉 〉 ) |
19 |
17 18
|
eqeqan12d |
⊢ ( ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ∧ ( 𝑎 ∈ ω ∧ 𝑏 ∈ ω ) ) → ( ( 𝐴 ∈𝑔 𝐵 ) = ( 𝑎 ∈𝑔 𝑏 ) ↔ 〈 ∅ , 〈 𝐴 , 𝐵 〉 〉 = 〈 ∅ , 〈 𝑎 , 𝑏 〉 〉 ) ) |
20 |
19
|
2rexbidva |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( ∃ 𝑎 ∈ ω ∃ 𝑏 ∈ ω ( 𝐴 ∈𝑔 𝐵 ) = ( 𝑎 ∈𝑔 𝑏 ) ↔ ∃ 𝑎 ∈ ω ∃ 𝑏 ∈ ω 〈 ∅ , 〈 𝐴 , 𝐵 〉 〉 = 〈 ∅ , 〈 𝑎 , 𝑏 〉 〉 ) ) |
21 |
16 20
|
mpbird |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ∃ 𝑎 ∈ ω ∃ 𝑏 ∈ ω ( 𝐴 ∈𝑔 𝐵 ) = ( 𝑎 ∈𝑔 𝑏 ) ) |
22 |
|
eqeq1 |
⊢ ( 𝑥 = ( 𝐴 ∈𝑔 𝐵 ) → ( 𝑥 = ( 𝑎 ∈𝑔 𝑏 ) ↔ ( 𝐴 ∈𝑔 𝐵 ) = ( 𝑎 ∈𝑔 𝑏 ) ) ) |
23 |
22
|
2rexbidv |
⊢ ( 𝑥 = ( 𝐴 ∈𝑔 𝐵 ) → ( ∃ 𝑎 ∈ ω ∃ 𝑏 ∈ ω 𝑥 = ( 𝑎 ∈𝑔 𝑏 ) ↔ ∃ 𝑎 ∈ ω ∃ 𝑏 ∈ ω ( 𝐴 ∈𝑔 𝐵 ) = ( 𝑎 ∈𝑔 𝑏 ) ) ) |
24 |
|
fmla0 |
⊢ ( Fmla ‘ ∅ ) = { 𝑥 ∈ V ∣ ∃ 𝑎 ∈ ω ∃ 𝑏 ∈ ω 𝑥 = ( 𝑎 ∈𝑔 𝑏 ) } |
25 |
23 24
|
elrab2 |
⊢ ( ( 𝐴 ∈𝑔 𝐵 ) ∈ ( Fmla ‘ ∅ ) ↔ ( ( 𝐴 ∈𝑔 𝐵 ) ∈ V ∧ ∃ 𝑎 ∈ ω ∃ 𝑏 ∈ ω ( 𝐴 ∈𝑔 𝐵 ) = ( 𝑎 ∈𝑔 𝑏 ) ) ) |
26 |
2 21 25
|
sylanbrc |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 ∈𝑔 𝐵 ) ∈ ( Fmla ‘ ∅ ) ) |
27 |
|
satefvfmla0 |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ ( 𝐴 ∈𝑔 𝐵 ) ∈ ( Fmla ‘ ∅ ) ) → ( 𝑀 Sat∈ ( 𝐴 ∈𝑔 𝐵 ) ) = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑎 ‘ ( 1st ‘ ( 2nd ‘ ( 𝐴 ∈𝑔 𝐵 ) ) ) ) ∈ ( 𝑎 ‘ ( 2nd ‘ ( 2nd ‘ ( 𝐴 ∈𝑔 𝐵 ) ) ) ) } ) |
28 |
26 27
|
sylan2 |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( 𝑀 Sat∈ ( 𝐴 ∈𝑔 𝐵 ) ) = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑎 ‘ ( 1st ‘ ( 2nd ‘ ( 𝐴 ∈𝑔 𝐵 ) ) ) ) ∈ ( 𝑎 ‘ ( 2nd ‘ ( 2nd ‘ ( 𝐴 ∈𝑔 𝐵 ) ) ) ) } ) |
29 |
1 28
|
syl5eq |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → 𝐸 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑎 ‘ ( 1st ‘ ( 2nd ‘ ( 𝐴 ∈𝑔 𝐵 ) ) ) ) ∈ ( 𝑎 ‘ ( 2nd ‘ ( 2nd ‘ ( 𝐴 ∈𝑔 𝐵 ) ) ) ) } ) |
30 |
29
|
eleq2d |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( 𝑆 ∈ 𝐸 ↔ 𝑆 ∈ { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑎 ‘ ( 1st ‘ ( 2nd ‘ ( 𝐴 ∈𝑔 𝐵 ) ) ) ) ∈ ( 𝑎 ‘ ( 2nd ‘ ( 2nd ‘ ( 𝐴 ∈𝑔 𝐵 ) ) ) ) } ) ) |
31 |
|
fveq1 |
⊢ ( 𝑎 = 𝑆 → ( 𝑎 ‘ ( 1st ‘ ( 2nd ‘ ( 𝐴 ∈𝑔 𝐵 ) ) ) ) = ( 𝑆 ‘ ( 1st ‘ ( 2nd ‘ ( 𝐴 ∈𝑔 𝐵 ) ) ) ) ) |
32 |
|
fveq1 |
⊢ ( 𝑎 = 𝑆 → ( 𝑎 ‘ ( 2nd ‘ ( 2nd ‘ ( 𝐴 ∈𝑔 𝐵 ) ) ) ) = ( 𝑆 ‘ ( 2nd ‘ ( 2nd ‘ ( 𝐴 ∈𝑔 𝐵 ) ) ) ) ) |
33 |
31 32
|
eleq12d |
⊢ ( 𝑎 = 𝑆 → ( ( 𝑎 ‘ ( 1st ‘ ( 2nd ‘ ( 𝐴 ∈𝑔 𝐵 ) ) ) ) ∈ ( 𝑎 ‘ ( 2nd ‘ ( 2nd ‘ ( 𝐴 ∈𝑔 𝐵 ) ) ) ) ↔ ( 𝑆 ‘ ( 1st ‘ ( 2nd ‘ ( 𝐴 ∈𝑔 𝐵 ) ) ) ) ∈ ( 𝑆 ‘ ( 2nd ‘ ( 2nd ‘ ( 𝐴 ∈𝑔 𝐵 ) ) ) ) ) ) |
34 |
33
|
elrab |
⊢ ( 𝑆 ∈ { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ( 𝑎 ‘ ( 1st ‘ ( 2nd ‘ ( 𝐴 ∈𝑔 𝐵 ) ) ) ) ∈ ( 𝑎 ‘ ( 2nd ‘ ( 2nd ‘ ( 𝐴 ∈𝑔 𝐵 ) ) ) ) } ↔ ( 𝑆 ∈ ( 𝑀 ↑m ω ) ∧ ( 𝑆 ‘ ( 1st ‘ ( 2nd ‘ ( 𝐴 ∈𝑔 𝐵 ) ) ) ) ∈ ( 𝑆 ‘ ( 2nd ‘ ( 2nd ‘ ( 𝐴 ∈𝑔 𝐵 ) ) ) ) ) ) |
35 |
30 34
|
bitrdi |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( 𝑆 ∈ 𝐸 ↔ ( 𝑆 ∈ ( 𝑀 ↑m ω ) ∧ ( 𝑆 ‘ ( 1st ‘ ( 2nd ‘ ( 𝐴 ∈𝑔 𝐵 ) ) ) ) ∈ ( 𝑆 ‘ ( 2nd ‘ ( 2nd ‘ ( 𝐴 ∈𝑔 𝐵 ) ) ) ) ) ) ) |
36 |
17
|
fveq2d |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 2nd ‘ ( 𝐴 ∈𝑔 𝐵 ) ) = ( 2nd ‘ 〈 ∅ , 〈 𝐴 , 𝐵 〉 〉 ) ) |
37 |
36
|
fveq2d |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 1st ‘ ( 2nd ‘ ( 𝐴 ∈𝑔 𝐵 ) ) ) = ( 1st ‘ ( 2nd ‘ 〈 ∅ , 〈 𝐴 , 𝐵 〉 〉 ) ) ) |
38 |
|
0ex |
⊢ ∅ ∈ V |
39 |
|
opex |
⊢ 〈 𝐴 , 𝐵 〉 ∈ V |
40 |
38 39
|
op2nd |
⊢ ( 2nd ‘ 〈 ∅ , 〈 𝐴 , 𝐵 〉 〉 ) = 〈 𝐴 , 𝐵 〉 |
41 |
40
|
fveq2i |
⊢ ( 1st ‘ ( 2nd ‘ 〈 ∅ , 〈 𝐴 , 𝐵 〉 〉 ) ) = ( 1st ‘ 〈 𝐴 , 𝐵 〉 ) |
42 |
|
op1stg |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 1st ‘ 〈 𝐴 , 𝐵 〉 ) = 𝐴 ) |
43 |
41 42
|
syl5eq |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 1st ‘ ( 2nd ‘ 〈 ∅ , 〈 𝐴 , 𝐵 〉 〉 ) ) = 𝐴 ) |
44 |
37 43
|
eqtrd |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 1st ‘ ( 2nd ‘ ( 𝐴 ∈𝑔 𝐵 ) ) ) = 𝐴 ) |
45 |
44
|
fveq2d |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝑆 ‘ ( 1st ‘ ( 2nd ‘ ( 𝐴 ∈𝑔 𝐵 ) ) ) ) = ( 𝑆 ‘ 𝐴 ) ) |
46 |
36
|
fveq2d |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 2nd ‘ ( 2nd ‘ ( 𝐴 ∈𝑔 𝐵 ) ) ) = ( 2nd ‘ ( 2nd ‘ 〈 ∅ , 〈 𝐴 , 𝐵 〉 〉 ) ) ) |
47 |
40
|
fveq2i |
⊢ ( 2nd ‘ ( 2nd ‘ 〈 ∅ , 〈 𝐴 , 𝐵 〉 〉 ) ) = ( 2nd ‘ 〈 𝐴 , 𝐵 〉 ) |
48 |
|
op2ndg |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 2nd ‘ 〈 𝐴 , 𝐵 〉 ) = 𝐵 ) |
49 |
47 48
|
syl5eq |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 2nd ‘ ( 2nd ‘ 〈 ∅ , 〈 𝐴 , 𝐵 〉 〉 ) ) = 𝐵 ) |
50 |
46 49
|
eqtrd |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 2nd ‘ ( 2nd ‘ ( 𝐴 ∈𝑔 𝐵 ) ) ) = 𝐵 ) |
51 |
50
|
fveq2d |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝑆 ‘ ( 2nd ‘ ( 2nd ‘ ( 𝐴 ∈𝑔 𝐵 ) ) ) ) = ( 𝑆 ‘ 𝐵 ) ) |
52 |
45 51
|
eleq12d |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( ( 𝑆 ‘ ( 1st ‘ ( 2nd ‘ ( 𝐴 ∈𝑔 𝐵 ) ) ) ) ∈ ( 𝑆 ‘ ( 2nd ‘ ( 2nd ‘ ( 𝐴 ∈𝑔 𝐵 ) ) ) ) ↔ ( 𝑆 ‘ 𝐴 ) ∈ ( 𝑆 ‘ 𝐵 ) ) ) |
53 |
52
|
adantl |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( ( 𝑆 ‘ ( 1st ‘ ( 2nd ‘ ( 𝐴 ∈𝑔 𝐵 ) ) ) ) ∈ ( 𝑆 ‘ ( 2nd ‘ ( 2nd ‘ ( 𝐴 ∈𝑔 𝐵 ) ) ) ) ↔ ( 𝑆 ‘ 𝐴 ) ∈ ( 𝑆 ‘ 𝐵 ) ) ) |
54 |
53
|
anbi2d |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( ( 𝑆 ∈ ( 𝑀 ↑m ω ) ∧ ( 𝑆 ‘ ( 1st ‘ ( 2nd ‘ ( 𝐴 ∈𝑔 𝐵 ) ) ) ) ∈ ( 𝑆 ‘ ( 2nd ‘ ( 2nd ‘ ( 𝐴 ∈𝑔 𝐵 ) ) ) ) ) ↔ ( 𝑆 ∈ ( 𝑀 ↑m ω ) ∧ ( 𝑆 ‘ 𝐴 ) ∈ ( 𝑆 ‘ 𝐵 ) ) ) ) |
55 |
35 54
|
bitrd |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( 𝑆 ∈ 𝐸 ↔ ( 𝑆 ∈ ( 𝑀 ↑m ω ) ∧ ( 𝑆 ‘ 𝐴 ) ∈ ( 𝑆 ‘ 𝐵 ) ) ) ) |