Step |
Hyp |
Ref |
Expression |
1 |
|
ordtval.1 |
⊢ 𝑋 = dom 𝑅 |
2 |
|
ordtval.2 |
⊢ 𝐴 = ran ( 𝑥 ∈ 𝑋 ↦ { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑥 } ) |
3 |
|
ordtval.3 |
⊢ 𝐵 = ran ( 𝑥 ∈ 𝑋 ↦ { 𝑦 ∈ 𝑋 ∣ ¬ 𝑥 𝑅 𝑦 } ) |
4 |
|
ordtval.4 |
⊢ 𝐶 = ran ( 𝑎 ∈ 𝑋 , 𝑏 ∈ 𝑋 ↦ { 𝑦 ∈ 𝑋 ∣ ( ¬ 𝑦 𝑅 𝑎 ∧ ¬ 𝑏 𝑅 𝑦 ) } ) |
5 |
|
ssun1 |
⊢ 𝐴 ⊆ ( 𝐴 ∪ 𝐵 ) |
6 |
|
ssun2 |
⊢ ( 𝐴 ∪ 𝐵 ) ⊆ ( { 𝑋 } ∪ ( 𝐴 ∪ 𝐵 ) ) |
7 |
1 2 3
|
ordtuni |
⊢ ( 𝑅 ∈ TosetRel → 𝑋 = ∪ ( { 𝑋 } ∪ ( 𝐴 ∪ 𝐵 ) ) ) |
8 |
|
dmexg |
⊢ ( 𝑅 ∈ TosetRel → dom 𝑅 ∈ V ) |
9 |
1 8
|
eqeltrid |
⊢ ( 𝑅 ∈ TosetRel → 𝑋 ∈ V ) |
10 |
7 9
|
eqeltrrd |
⊢ ( 𝑅 ∈ TosetRel → ∪ ( { 𝑋 } ∪ ( 𝐴 ∪ 𝐵 ) ) ∈ V ) |
11 |
|
uniexb |
⊢ ( ( { 𝑋 } ∪ ( 𝐴 ∪ 𝐵 ) ) ∈ V ↔ ∪ ( { 𝑋 } ∪ ( 𝐴 ∪ 𝐵 ) ) ∈ V ) |
12 |
10 11
|
sylibr |
⊢ ( 𝑅 ∈ TosetRel → ( { 𝑋 } ∪ ( 𝐴 ∪ 𝐵 ) ) ∈ V ) |
13 |
|
ssexg |
⊢ ( ( ( 𝐴 ∪ 𝐵 ) ⊆ ( { 𝑋 } ∪ ( 𝐴 ∪ 𝐵 ) ) ∧ ( { 𝑋 } ∪ ( 𝐴 ∪ 𝐵 ) ) ∈ V ) → ( 𝐴 ∪ 𝐵 ) ∈ V ) |
14 |
6 12 13
|
sylancr |
⊢ ( 𝑅 ∈ TosetRel → ( 𝐴 ∪ 𝐵 ) ∈ V ) |
15 |
|
ssexg |
⊢ ( ( 𝐴 ⊆ ( 𝐴 ∪ 𝐵 ) ∧ ( 𝐴 ∪ 𝐵 ) ∈ V ) → 𝐴 ∈ V ) |
16 |
5 14 15
|
sylancr |
⊢ ( 𝑅 ∈ TosetRel → 𝐴 ∈ V ) |
17 |
|
ssun2 |
⊢ 𝐵 ⊆ ( 𝐴 ∪ 𝐵 ) |
18 |
|
ssexg |
⊢ ( ( 𝐵 ⊆ ( 𝐴 ∪ 𝐵 ) ∧ ( 𝐴 ∪ 𝐵 ) ∈ V ) → 𝐵 ∈ V ) |
19 |
17 14 18
|
sylancr |
⊢ ( 𝑅 ∈ TosetRel → 𝐵 ∈ V ) |
20 |
|
elfiun |
⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( 𝑧 ∈ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) ↔ ( 𝑧 ∈ ( fi ‘ 𝐴 ) ∨ 𝑧 ∈ ( fi ‘ 𝐵 ) ∨ ∃ 𝑚 ∈ ( fi ‘ 𝐴 ) ∃ 𝑛 ∈ ( fi ‘ 𝐵 ) 𝑧 = ( 𝑚 ∩ 𝑛 ) ) ) ) |
21 |
16 19 20
|
syl2anc |
⊢ ( 𝑅 ∈ TosetRel → ( 𝑧 ∈ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) ↔ ( 𝑧 ∈ ( fi ‘ 𝐴 ) ∨ 𝑧 ∈ ( fi ‘ 𝐵 ) ∨ ∃ 𝑚 ∈ ( fi ‘ 𝐴 ) ∃ 𝑛 ∈ ( fi ‘ 𝐵 ) 𝑧 = ( 𝑚 ∩ 𝑛 ) ) ) ) |
22 |
1 2
|
ordtbaslem |
⊢ ( 𝑅 ∈ TosetRel → ( fi ‘ 𝐴 ) = 𝐴 ) |
23 |
22 5
|
eqsstrdi |
⊢ ( 𝑅 ∈ TosetRel → ( fi ‘ 𝐴 ) ⊆ ( 𝐴 ∪ 𝐵 ) ) |
24 |
|
ssun1 |
⊢ ( 𝐴 ∪ 𝐵 ) ⊆ ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) |
25 |
23 24
|
sstrdi |
⊢ ( 𝑅 ∈ TosetRel → ( fi ‘ 𝐴 ) ⊆ ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) ) |
26 |
25
|
sseld |
⊢ ( 𝑅 ∈ TosetRel → ( 𝑧 ∈ ( fi ‘ 𝐴 ) → 𝑧 ∈ ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) ) ) |
27 |
|
cnvtsr |
⊢ ( 𝑅 ∈ TosetRel → ◡ 𝑅 ∈ TosetRel ) |
28 |
|
df-rn |
⊢ ran 𝑅 = dom ◡ 𝑅 |
29 |
|
eqid |
⊢ ran ( 𝑥 ∈ ran 𝑅 ↦ { 𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦 ◡ 𝑅 𝑥 } ) = ran ( 𝑥 ∈ ran 𝑅 ↦ { 𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦 ◡ 𝑅 𝑥 } ) |
30 |
28 29
|
ordtbaslem |
⊢ ( ◡ 𝑅 ∈ TosetRel → ( fi ‘ ran ( 𝑥 ∈ ran 𝑅 ↦ { 𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦 ◡ 𝑅 𝑥 } ) ) = ran ( 𝑥 ∈ ran 𝑅 ↦ { 𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦 ◡ 𝑅 𝑥 } ) ) |
31 |
27 30
|
syl |
⊢ ( 𝑅 ∈ TosetRel → ( fi ‘ ran ( 𝑥 ∈ ran 𝑅 ↦ { 𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦 ◡ 𝑅 𝑥 } ) ) = ran ( 𝑥 ∈ ran 𝑅 ↦ { 𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦 ◡ 𝑅 𝑥 } ) ) |
32 |
|
tsrps |
⊢ ( 𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel ) |
33 |
1
|
psrn |
⊢ ( 𝑅 ∈ PosetRel → 𝑋 = ran 𝑅 ) |
34 |
32 33
|
syl |
⊢ ( 𝑅 ∈ TosetRel → 𝑋 = ran 𝑅 ) |
35 |
|
vex |
⊢ 𝑦 ∈ V |
36 |
|
vex |
⊢ 𝑥 ∈ V |
37 |
35 36
|
brcnv |
⊢ ( 𝑦 ◡ 𝑅 𝑥 ↔ 𝑥 𝑅 𝑦 ) |
38 |
37
|
bicomi |
⊢ ( 𝑥 𝑅 𝑦 ↔ 𝑦 ◡ 𝑅 𝑥 ) |
39 |
38
|
notbii |
⊢ ( ¬ 𝑥 𝑅 𝑦 ↔ ¬ 𝑦 ◡ 𝑅 𝑥 ) |
40 |
39
|
a1i |
⊢ ( 𝑅 ∈ TosetRel → ( ¬ 𝑥 𝑅 𝑦 ↔ ¬ 𝑦 ◡ 𝑅 𝑥 ) ) |
41 |
34 40
|
rabeqbidv |
⊢ ( 𝑅 ∈ TosetRel → { 𝑦 ∈ 𝑋 ∣ ¬ 𝑥 𝑅 𝑦 } = { 𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦 ◡ 𝑅 𝑥 } ) |
42 |
34 41
|
mpteq12dv |
⊢ ( 𝑅 ∈ TosetRel → ( 𝑥 ∈ 𝑋 ↦ { 𝑦 ∈ 𝑋 ∣ ¬ 𝑥 𝑅 𝑦 } ) = ( 𝑥 ∈ ran 𝑅 ↦ { 𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦 ◡ 𝑅 𝑥 } ) ) |
43 |
42
|
rneqd |
⊢ ( 𝑅 ∈ TosetRel → ran ( 𝑥 ∈ 𝑋 ↦ { 𝑦 ∈ 𝑋 ∣ ¬ 𝑥 𝑅 𝑦 } ) = ran ( 𝑥 ∈ ran 𝑅 ↦ { 𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦 ◡ 𝑅 𝑥 } ) ) |
44 |
3 43
|
syl5eq |
⊢ ( 𝑅 ∈ TosetRel → 𝐵 = ran ( 𝑥 ∈ ran 𝑅 ↦ { 𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦 ◡ 𝑅 𝑥 } ) ) |
45 |
44
|
fveq2d |
⊢ ( 𝑅 ∈ TosetRel → ( fi ‘ 𝐵 ) = ( fi ‘ ran ( 𝑥 ∈ ran 𝑅 ↦ { 𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦 ◡ 𝑅 𝑥 } ) ) ) |
46 |
31 45 44
|
3eqtr4d |
⊢ ( 𝑅 ∈ TosetRel → ( fi ‘ 𝐵 ) = 𝐵 ) |
47 |
46 17
|
eqsstrdi |
⊢ ( 𝑅 ∈ TosetRel → ( fi ‘ 𝐵 ) ⊆ ( 𝐴 ∪ 𝐵 ) ) |
48 |
47 24
|
sstrdi |
⊢ ( 𝑅 ∈ TosetRel → ( fi ‘ 𝐵 ) ⊆ ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) ) |
49 |
48
|
sseld |
⊢ ( 𝑅 ∈ TosetRel → ( 𝑧 ∈ ( fi ‘ 𝐵 ) → 𝑧 ∈ ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) ) ) |
50 |
|
ssun2 |
⊢ 𝐶 ⊆ ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) |
51 |
22 2
|
eqtrdi |
⊢ ( 𝑅 ∈ TosetRel → ( fi ‘ 𝐴 ) = ran ( 𝑥 ∈ 𝑋 ↦ { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑥 } ) ) |
52 |
51
|
eleq2d |
⊢ ( 𝑅 ∈ TosetRel → ( 𝑚 ∈ ( fi ‘ 𝐴 ) ↔ 𝑚 ∈ ran ( 𝑥 ∈ 𝑋 ↦ { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑥 } ) ) ) |
53 |
|
breq2 |
⊢ ( 𝑥 = 𝑎 → ( 𝑦 𝑅 𝑥 ↔ 𝑦 𝑅 𝑎 ) ) |
54 |
53
|
notbid |
⊢ ( 𝑥 = 𝑎 → ( ¬ 𝑦 𝑅 𝑥 ↔ ¬ 𝑦 𝑅 𝑎 ) ) |
55 |
54
|
rabbidv |
⊢ ( 𝑥 = 𝑎 → { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑥 } = { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑎 } ) |
56 |
55
|
cbvmptv |
⊢ ( 𝑥 ∈ 𝑋 ↦ { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑥 } ) = ( 𝑎 ∈ 𝑋 ↦ { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑎 } ) |
57 |
56
|
elrnmpt |
⊢ ( 𝑚 ∈ V → ( 𝑚 ∈ ran ( 𝑥 ∈ 𝑋 ↦ { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑥 } ) ↔ ∃ 𝑎 ∈ 𝑋 𝑚 = { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑎 } ) ) |
58 |
57
|
elv |
⊢ ( 𝑚 ∈ ran ( 𝑥 ∈ 𝑋 ↦ { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑥 } ) ↔ ∃ 𝑎 ∈ 𝑋 𝑚 = { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑎 } ) |
59 |
52 58
|
bitrdi |
⊢ ( 𝑅 ∈ TosetRel → ( 𝑚 ∈ ( fi ‘ 𝐴 ) ↔ ∃ 𝑎 ∈ 𝑋 𝑚 = { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑎 } ) ) |
60 |
46 3
|
eqtrdi |
⊢ ( 𝑅 ∈ TosetRel → ( fi ‘ 𝐵 ) = ran ( 𝑥 ∈ 𝑋 ↦ { 𝑦 ∈ 𝑋 ∣ ¬ 𝑥 𝑅 𝑦 } ) ) |
61 |
60
|
eleq2d |
⊢ ( 𝑅 ∈ TosetRel → ( 𝑛 ∈ ( fi ‘ 𝐵 ) ↔ 𝑛 ∈ ran ( 𝑥 ∈ 𝑋 ↦ { 𝑦 ∈ 𝑋 ∣ ¬ 𝑥 𝑅 𝑦 } ) ) ) |
62 |
|
breq1 |
⊢ ( 𝑥 = 𝑏 → ( 𝑥 𝑅 𝑦 ↔ 𝑏 𝑅 𝑦 ) ) |
63 |
62
|
notbid |
⊢ ( 𝑥 = 𝑏 → ( ¬ 𝑥 𝑅 𝑦 ↔ ¬ 𝑏 𝑅 𝑦 ) ) |
64 |
63
|
rabbidv |
⊢ ( 𝑥 = 𝑏 → { 𝑦 ∈ 𝑋 ∣ ¬ 𝑥 𝑅 𝑦 } = { 𝑦 ∈ 𝑋 ∣ ¬ 𝑏 𝑅 𝑦 } ) |
65 |
64
|
cbvmptv |
⊢ ( 𝑥 ∈ 𝑋 ↦ { 𝑦 ∈ 𝑋 ∣ ¬ 𝑥 𝑅 𝑦 } ) = ( 𝑏 ∈ 𝑋 ↦ { 𝑦 ∈ 𝑋 ∣ ¬ 𝑏 𝑅 𝑦 } ) |
66 |
65
|
elrnmpt |
⊢ ( 𝑛 ∈ V → ( 𝑛 ∈ ran ( 𝑥 ∈ 𝑋 ↦ { 𝑦 ∈ 𝑋 ∣ ¬ 𝑥 𝑅 𝑦 } ) ↔ ∃ 𝑏 ∈ 𝑋 𝑛 = { 𝑦 ∈ 𝑋 ∣ ¬ 𝑏 𝑅 𝑦 } ) ) |
67 |
66
|
elv |
⊢ ( 𝑛 ∈ ran ( 𝑥 ∈ 𝑋 ↦ { 𝑦 ∈ 𝑋 ∣ ¬ 𝑥 𝑅 𝑦 } ) ↔ ∃ 𝑏 ∈ 𝑋 𝑛 = { 𝑦 ∈ 𝑋 ∣ ¬ 𝑏 𝑅 𝑦 } ) |
68 |
61 67
|
bitrdi |
⊢ ( 𝑅 ∈ TosetRel → ( 𝑛 ∈ ( fi ‘ 𝐵 ) ↔ ∃ 𝑏 ∈ 𝑋 𝑛 = { 𝑦 ∈ 𝑋 ∣ ¬ 𝑏 𝑅 𝑦 } ) ) |
69 |
59 68
|
anbi12d |
⊢ ( 𝑅 ∈ TosetRel → ( ( 𝑚 ∈ ( fi ‘ 𝐴 ) ∧ 𝑛 ∈ ( fi ‘ 𝐵 ) ) ↔ ( ∃ 𝑎 ∈ 𝑋 𝑚 = { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑎 } ∧ ∃ 𝑏 ∈ 𝑋 𝑛 = { 𝑦 ∈ 𝑋 ∣ ¬ 𝑏 𝑅 𝑦 } ) ) ) |
70 |
|
reeanv |
⊢ ( ∃ 𝑎 ∈ 𝑋 ∃ 𝑏 ∈ 𝑋 ( 𝑚 = { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑎 } ∧ 𝑛 = { 𝑦 ∈ 𝑋 ∣ ¬ 𝑏 𝑅 𝑦 } ) ↔ ( ∃ 𝑎 ∈ 𝑋 𝑚 = { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑎 } ∧ ∃ 𝑏 ∈ 𝑋 𝑛 = { 𝑦 ∈ 𝑋 ∣ ¬ 𝑏 𝑅 𝑦 } ) ) |
71 |
|
ineq12 |
⊢ ( ( 𝑚 = { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑎 } ∧ 𝑛 = { 𝑦 ∈ 𝑋 ∣ ¬ 𝑏 𝑅 𝑦 } ) → ( 𝑚 ∩ 𝑛 ) = ( { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑎 } ∩ { 𝑦 ∈ 𝑋 ∣ ¬ 𝑏 𝑅 𝑦 } ) ) |
72 |
|
inrab |
⊢ ( { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑎 } ∩ { 𝑦 ∈ 𝑋 ∣ ¬ 𝑏 𝑅 𝑦 } ) = { 𝑦 ∈ 𝑋 ∣ ( ¬ 𝑦 𝑅 𝑎 ∧ ¬ 𝑏 𝑅 𝑦 ) } |
73 |
71 72
|
eqtrdi |
⊢ ( ( 𝑚 = { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑎 } ∧ 𝑛 = { 𝑦 ∈ 𝑋 ∣ ¬ 𝑏 𝑅 𝑦 } ) → ( 𝑚 ∩ 𝑛 ) = { 𝑦 ∈ 𝑋 ∣ ( ¬ 𝑦 𝑅 𝑎 ∧ ¬ 𝑏 𝑅 𝑦 ) } ) |
74 |
73
|
reximi |
⊢ ( ∃ 𝑏 ∈ 𝑋 ( 𝑚 = { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑎 } ∧ 𝑛 = { 𝑦 ∈ 𝑋 ∣ ¬ 𝑏 𝑅 𝑦 } ) → ∃ 𝑏 ∈ 𝑋 ( 𝑚 ∩ 𝑛 ) = { 𝑦 ∈ 𝑋 ∣ ( ¬ 𝑦 𝑅 𝑎 ∧ ¬ 𝑏 𝑅 𝑦 ) } ) |
75 |
74
|
reximi |
⊢ ( ∃ 𝑎 ∈ 𝑋 ∃ 𝑏 ∈ 𝑋 ( 𝑚 = { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑎 } ∧ 𝑛 = { 𝑦 ∈ 𝑋 ∣ ¬ 𝑏 𝑅 𝑦 } ) → ∃ 𝑎 ∈ 𝑋 ∃ 𝑏 ∈ 𝑋 ( 𝑚 ∩ 𝑛 ) = { 𝑦 ∈ 𝑋 ∣ ( ¬ 𝑦 𝑅 𝑎 ∧ ¬ 𝑏 𝑅 𝑦 ) } ) |
76 |
70 75
|
sylbir |
⊢ ( ( ∃ 𝑎 ∈ 𝑋 𝑚 = { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑎 } ∧ ∃ 𝑏 ∈ 𝑋 𝑛 = { 𝑦 ∈ 𝑋 ∣ ¬ 𝑏 𝑅 𝑦 } ) → ∃ 𝑎 ∈ 𝑋 ∃ 𝑏 ∈ 𝑋 ( 𝑚 ∩ 𝑛 ) = { 𝑦 ∈ 𝑋 ∣ ( ¬ 𝑦 𝑅 𝑎 ∧ ¬ 𝑏 𝑅 𝑦 ) } ) |
77 |
69 76
|
syl6bi |
⊢ ( 𝑅 ∈ TosetRel → ( ( 𝑚 ∈ ( fi ‘ 𝐴 ) ∧ 𝑛 ∈ ( fi ‘ 𝐵 ) ) → ∃ 𝑎 ∈ 𝑋 ∃ 𝑏 ∈ 𝑋 ( 𝑚 ∩ 𝑛 ) = { 𝑦 ∈ 𝑋 ∣ ( ¬ 𝑦 𝑅 𝑎 ∧ ¬ 𝑏 𝑅 𝑦 ) } ) ) |
78 |
77
|
imp |
⊢ ( ( 𝑅 ∈ TosetRel ∧ ( 𝑚 ∈ ( fi ‘ 𝐴 ) ∧ 𝑛 ∈ ( fi ‘ 𝐵 ) ) ) → ∃ 𝑎 ∈ 𝑋 ∃ 𝑏 ∈ 𝑋 ( 𝑚 ∩ 𝑛 ) = { 𝑦 ∈ 𝑋 ∣ ( ¬ 𝑦 𝑅 𝑎 ∧ ¬ 𝑏 𝑅 𝑦 ) } ) |
79 |
|
vex |
⊢ 𝑚 ∈ V |
80 |
79
|
inex1 |
⊢ ( 𝑚 ∩ 𝑛 ) ∈ V |
81 |
|
eqid |
⊢ ( 𝑎 ∈ 𝑋 , 𝑏 ∈ 𝑋 ↦ { 𝑦 ∈ 𝑋 ∣ ( ¬ 𝑦 𝑅 𝑎 ∧ ¬ 𝑏 𝑅 𝑦 ) } ) = ( 𝑎 ∈ 𝑋 , 𝑏 ∈ 𝑋 ↦ { 𝑦 ∈ 𝑋 ∣ ( ¬ 𝑦 𝑅 𝑎 ∧ ¬ 𝑏 𝑅 𝑦 ) } ) |
82 |
81
|
elrnmpog |
⊢ ( ( 𝑚 ∩ 𝑛 ) ∈ V → ( ( 𝑚 ∩ 𝑛 ) ∈ ran ( 𝑎 ∈ 𝑋 , 𝑏 ∈ 𝑋 ↦ { 𝑦 ∈ 𝑋 ∣ ( ¬ 𝑦 𝑅 𝑎 ∧ ¬ 𝑏 𝑅 𝑦 ) } ) ↔ ∃ 𝑎 ∈ 𝑋 ∃ 𝑏 ∈ 𝑋 ( 𝑚 ∩ 𝑛 ) = { 𝑦 ∈ 𝑋 ∣ ( ¬ 𝑦 𝑅 𝑎 ∧ ¬ 𝑏 𝑅 𝑦 ) } ) ) |
83 |
80 82
|
ax-mp |
⊢ ( ( 𝑚 ∩ 𝑛 ) ∈ ran ( 𝑎 ∈ 𝑋 , 𝑏 ∈ 𝑋 ↦ { 𝑦 ∈ 𝑋 ∣ ( ¬ 𝑦 𝑅 𝑎 ∧ ¬ 𝑏 𝑅 𝑦 ) } ) ↔ ∃ 𝑎 ∈ 𝑋 ∃ 𝑏 ∈ 𝑋 ( 𝑚 ∩ 𝑛 ) = { 𝑦 ∈ 𝑋 ∣ ( ¬ 𝑦 𝑅 𝑎 ∧ ¬ 𝑏 𝑅 𝑦 ) } ) |
84 |
78 83
|
sylibr |
⊢ ( ( 𝑅 ∈ TosetRel ∧ ( 𝑚 ∈ ( fi ‘ 𝐴 ) ∧ 𝑛 ∈ ( fi ‘ 𝐵 ) ) ) → ( 𝑚 ∩ 𝑛 ) ∈ ran ( 𝑎 ∈ 𝑋 , 𝑏 ∈ 𝑋 ↦ { 𝑦 ∈ 𝑋 ∣ ( ¬ 𝑦 𝑅 𝑎 ∧ ¬ 𝑏 𝑅 𝑦 ) } ) ) |
85 |
84 4
|
eleqtrrdi |
⊢ ( ( 𝑅 ∈ TosetRel ∧ ( 𝑚 ∈ ( fi ‘ 𝐴 ) ∧ 𝑛 ∈ ( fi ‘ 𝐵 ) ) ) → ( 𝑚 ∩ 𝑛 ) ∈ 𝐶 ) |
86 |
50 85
|
sselid |
⊢ ( ( 𝑅 ∈ TosetRel ∧ ( 𝑚 ∈ ( fi ‘ 𝐴 ) ∧ 𝑛 ∈ ( fi ‘ 𝐵 ) ) ) → ( 𝑚 ∩ 𝑛 ) ∈ ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) ) |
87 |
|
eleq1 |
⊢ ( 𝑧 = ( 𝑚 ∩ 𝑛 ) → ( 𝑧 ∈ ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) ↔ ( 𝑚 ∩ 𝑛 ) ∈ ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) ) ) |
88 |
86 87
|
syl5ibrcom |
⊢ ( ( 𝑅 ∈ TosetRel ∧ ( 𝑚 ∈ ( fi ‘ 𝐴 ) ∧ 𝑛 ∈ ( fi ‘ 𝐵 ) ) ) → ( 𝑧 = ( 𝑚 ∩ 𝑛 ) → 𝑧 ∈ ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) ) ) |
89 |
88
|
rexlimdvva |
⊢ ( 𝑅 ∈ TosetRel → ( ∃ 𝑚 ∈ ( fi ‘ 𝐴 ) ∃ 𝑛 ∈ ( fi ‘ 𝐵 ) 𝑧 = ( 𝑚 ∩ 𝑛 ) → 𝑧 ∈ ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) ) ) |
90 |
26 49 89
|
3jaod |
⊢ ( 𝑅 ∈ TosetRel → ( ( 𝑧 ∈ ( fi ‘ 𝐴 ) ∨ 𝑧 ∈ ( fi ‘ 𝐵 ) ∨ ∃ 𝑚 ∈ ( fi ‘ 𝐴 ) ∃ 𝑛 ∈ ( fi ‘ 𝐵 ) 𝑧 = ( 𝑚 ∩ 𝑛 ) ) → 𝑧 ∈ ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) ) ) |
91 |
21 90
|
sylbid |
⊢ ( 𝑅 ∈ TosetRel → ( 𝑧 ∈ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) → 𝑧 ∈ ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) ) ) |
92 |
91
|
ssrdv |
⊢ ( 𝑅 ∈ TosetRel → ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) ⊆ ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) ) |
93 |
|
ssfii |
⊢ ( ( 𝐴 ∪ 𝐵 ) ∈ V → ( 𝐴 ∪ 𝐵 ) ⊆ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) ) |
94 |
14 93
|
syl |
⊢ ( 𝑅 ∈ TosetRel → ( 𝐴 ∪ 𝐵 ) ⊆ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) ) |
95 |
94
|
adantr |
⊢ ( ( 𝑅 ∈ TosetRel ∧ ( 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ) ) → ( 𝐴 ∪ 𝐵 ) ⊆ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) ) |
96 |
|
simprl |
⊢ ( ( 𝑅 ∈ TosetRel ∧ ( 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ) ) → 𝑎 ∈ 𝑋 ) |
97 |
|
eqidd |
⊢ ( ( 𝑅 ∈ TosetRel ∧ ( 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ) ) → { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑎 } = { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑎 } ) |
98 |
55
|
rspceeqv |
⊢ ( ( 𝑎 ∈ 𝑋 ∧ { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑎 } = { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑎 } ) → ∃ 𝑥 ∈ 𝑋 { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑎 } = { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑥 } ) |
99 |
96 97 98
|
syl2anc |
⊢ ( ( 𝑅 ∈ TosetRel ∧ ( 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ) ) → ∃ 𝑥 ∈ 𝑋 { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑎 } = { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑥 } ) |
100 |
9
|
adantr |
⊢ ( ( 𝑅 ∈ TosetRel ∧ ( 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ) ) → 𝑋 ∈ V ) |
101 |
|
rabexg |
⊢ ( 𝑋 ∈ V → { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑎 } ∈ V ) |
102 |
|
eqid |
⊢ ( 𝑥 ∈ 𝑋 ↦ { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑥 } ) = ( 𝑥 ∈ 𝑋 ↦ { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑥 } ) |
103 |
102
|
elrnmpt |
⊢ ( { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑎 } ∈ V → ( { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑎 } ∈ ran ( 𝑥 ∈ 𝑋 ↦ { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑥 } ) ↔ ∃ 𝑥 ∈ 𝑋 { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑎 } = { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑥 } ) ) |
104 |
100 101 103
|
3syl |
⊢ ( ( 𝑅 ∈ TosetRel ∧ ( 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ) ) → ( { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑎 } ∈ ran ( 𝑥 ∈ 𝑋 ↦ { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑥 } ) ↔ ∃ 𝑥 ∈ 𝑋 { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑎 } = { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑥 } ) ) |
105 |
99 104
|
mpbird |
⊢ ( ( 𝑅 ∈ TosetRel ∧ ( 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ) ) → { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑎 } ∈ ran ( 𝑥 ∈ 𝑋 ↦ { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑥 } ) ) |
106 |
105 2
|
eleqtrrdi |
⊢ ( ( 𝑅 ∈ TosetRel ∧ ( 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ) ) → { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑎 } ∈ 𝐴 ) |
107 |
5 106
|
sselid |
⊢ ( ( 𝑅 ∈ TosetRel ∧ ( 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ) ) → { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑎 } ∈ ( 𝐴 ∪ 𝐵 ) ) |
108 |
95 107
|
sseldd |
⊢ ( ( 𝑅 ∈ TosetRel ∧ ( 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ) ) → { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑎 } ∈ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) ) |
109 |
|
simprr |
⊢ ( ( 𝑅 ∈ TosetRel ∧ ( 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ) ) → 𝑏 ∈ 𝑋 ) |
110 |
|
eqidd |
⊢ ( ( 𝑅 ∈ TosetRel ∧ ( 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ) ) → { 𝑦 ∈ 𝑋 ∣ ¬ 𝑏 𝑅 𝑦 } = { 𝑦 ∈ 𝑋 ∣ ¬ 𝑏 𝑅 𝑦 } ) |
111 |
64
|
rspceeqv |
⊢ ( ( 𝑏 ∈ 𝑋 ∧ { 𝑦 ∈ 𝑋 ∣ ¬ 𝑏 𝑅 𝑦 } = { 𝑦 ∈ 𝑋 ∣ ¬ 𝑏 𝑅 𝑦 } ) → ∃ 𝑥 ∈ 𝑋 { 𝑦 ∈ 𝑋 ∣ ¬ 𝑏 𝑅 𝑦 } = { 𝑦 ∈ 𝑋 ∣ ¬ 𝑥 𝑅 𝑦 } ) |
112 |
109 110 111
|
syl2anc |
⊢ ( ( 𝑅 ∈ TosetRel ∧ ( 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ) ) → ∃ 𝑥 ∈ 𝑋 { 𝑦 ∈ 𝑋 ∣ ¬ 𝑏 𝑅 𝑦 } = { 𝑦 ∈ 𝑋 ∣ ¬ 𝑥 𝑅 𝑦 } ) |
113 |
|
rabexg |
⊢ ( 𝑋 ∈ V → { 𝑦 ∈ 𝑋 ∣ ¬ 𝑏 𝑅 𝑦 } ∈ V ) |
114 |
|
eqid |
⊢ ( 𝑥 ∈ 𝑋 ↦ { 𝑦 ∈ 𝑋 ∣ ¬ 𝑥 𝑅 𝑦 } ) = ( 𝑥 ∈ 𝑋 ↦ { 𝑦 ∈ 𝑋 ∣ ¬ 𝑥 𝑅 𝑦 } ) |
115 |
114
|
elrnmpt |
⊢ ( { 𝑦 ∈ 𝑋 ∣ ¬ 𝑏 𝑅 𝑦 } ∈ V → ( { 𝑦 ∈ 𝑋 ∣ ¬ 𝑏 𝑅 𝑦 } ∈ ran ( 𝑥 ∈ 𝑋 ↦ { 𝑦 ∈ 𝑋 ∣ ¬ 𝑥 𝑅 𝑦 } ) ↔ ∃ 𝑥 ∈ 𝑋 { 𝑦 ∈ 𝑋 ∣ ¬ 𝑏 𝑅 𝑦 } = { 𝑦 ∈ 𝑋 ∣ ¬ 𝑥 𝑅 𝑦 } ) ) |
116 |
100 113 115
|
3syl |
⊢ ( ( 𝑅 ∈ TosetRel ∧ ( 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ) ) → ( { 𝑦 ∈ 𝑋 ∣ ¬ 𝑏 𝑅 𝑦 } ∈ ran ( 𝑥 ∈ 𝑋 ↦ { 𝑦 ∈ 𝑋 ∣ ¬ 𝑥 𝑅 𝑦 } ) ↔ ∃ 𝑥 ∈ 𝑋 { 𝑦 ∈ 𝑋 ∣ ¬ 𝑏 𝑅 𝑦 } = { 𝑦 ∈ 𝑋 ∣ ¬ 𝑥 𝑅 𝑦 } ) ) |
117 |
112 116
|
mpbird |
⊢ ( ( 𝑅 ∈ TosetRel ∧ ( 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ) ) → { 𝑦 ∈ 𝑋 ∣ ¬ 𝑏 𝑅 𝑦 } ∈ ran ( 𝑥 ∈ 𝑋 ↦ { 𝑦 ∈ 𝑋 ∣ ¬ 𝑥 𝑅 𝑦 } ) ) |
118 |
117 3
|
eleqtrrdi |
⊢ ( ( 𝑅 ∈ TosetRel ∧ ( 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ) ) → { 𝑦 ∈ 𝑋 ∣ ¬ 𝑏 𝑅 𝑦 } ∈ 𝐵 ) |
119 |
17 118
|
sselid |
⊢ ( ( 𝑅 ∈ TosetRel ∧ ( 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ) ) → { 𝑦 ∈ 𝑋 ∣ ¬ 𝑏 𝑅 𝑦 } ∈ ( 𝐴 ∪ 𝐵 ) ) |
120 |
95 119
|
sseldd |
⊢ ( ( 𝑅 ∈ TosetRel ∧ ( 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ) ) → { 𝑦 ∈ 𝑋 ∣ ¬ 𝑏 𝑅 𝑦 } ∈ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) ) |
121 |
|
fiin |
⊢ ( ( { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑎 } ∈ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) ∧ { 𝑦 ∈ 𝑋 ∣ ¬ 𝑏 𝑅 𝑦 } ∈ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) ) → ( { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑎 } ∩ { 𝑦 ∈ 𝑋 ∣ ¬ 𝑏 𝑅 𝑦 } ) ∈ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) ) |
122 |
108 120 121
|
syl2anc |
⊢ ( ( 𝑅 ∈ TosetRel ∧ ( 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ) ) → ( { 𝑦 ∈ 𝑋 ∣ ¬ 𝑦 𝑅 𝑎 } ∩ { 𝑦 ∈ 𝑋 ∣ ¬ 𝑏 𝑅 𝑦 } ) ∈ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) ) |
123 |
72 122
|
eqeltrrid |
⊢ ( ( 𝑅 ∈ TosetRel ∧ ( 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ) ) → { 𝑦 ∈ 𝑋 ∣ ( ¬ 𝑦 𝑅 𝑎 ∧ ¬ 𝑏 𝑅 𝑦 ) } ∈ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) ) |
124 |
123
|
ralrimivva |
⊢ ( 𝑅 ∈ TosetRel → ∀ 𝑎 ∈ 𝑋 ∀ 𝑏 ∈ 𝑋 { 𝑦 ∈ 𝑋 ∣ ( ¬ 𝑦 𝑅 𝑎 ∧ ¬ 𝑏 𝑅 𝑦 ) } ∈ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) ) |
125 |
81
|
fmpo |
⊢ ( ∀ 𝑎 ∈ 𝑋 ∀ 𝑏 ∈ 𝑋 { 𝑦 ∈ 𝑋 ∣ ( ¬ 𝑦 𝑅 𝑎 ∧ ¬ 𝑏 𝑅 𝑦 ) } ∈ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) ↔ ( 𝑎 ∈ 𝑋 , 𝑏 ∈ 𝑋 ↦ { 𝑦 ∈ 𝑋 ∣ ( ¬ 𝑦 𝑅 𝑎 ∧ ¬ 𝑏 𝑅 𝑦 ) } ) : ( 𝑋 × 𝑋 ) ⟶ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) ) |
126 |
124 125
|
sylib |
⊢ ( 𝑅 ∈ TosetRel → ( 𝑎 ∈ 𝑋 , 𝑏 ∈ 𝑋 ↦ { 𝑦 ∈ 𝑋 ∣ ( ¬ 𝑦 𝑅 𝑎 ∧ ¬ 𝑏 𝑅 𝑦 ) } ) : ( 𝑋 × 𝑋 ) ⟶ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) ) |
127 |
126
|
frnd |
⊢ ( 𝑅 ∈ TosetRel → ran ( 𝑎 ∈ 𝑋 , 𝑏 ∈ 𝑋 ↦ { 𝑦 ∈ 𝑋 ∣ ( ¬ 𝑦 𝑅 𝑎 ∧ ¬ 𝑏 𝑅 𝑦 ) } ) ⊆ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) ) |
128 |
4 127
|
eqsstrid |
⊢ ( 𝑅 ∈ TosetRel → 𝐶 ⊆ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) ) |
129 |
94 128
|
unssd |
⊢ ( 𝑅 ∈ TosetRel → ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) ⊆ ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) ) |
130 |
92 129
|
eqssd |
⊢ ( 𝑅 ∈ TosetRel → ( fi ‘ ( 𝐴 ∪ 𝐵 ) ) = ( ( 𝐴 ∪ 𝐵 ) ∪ 𝐶 ) ) |