Step |
Hyp |
Ref |
Expression |
1 |
|
raleq |
⊢ ( 𝑦 = ∅ → ( ∀ 𝑘 ∈ 𝑦 𝐵 ∈ dom vol ↔ ∀ 𝑘 ∈ ∅ 𝐵 ∈ dom vol ) ) |
2 |
|
iuneq1 |
⊢ ( 𝑦 = ∅ → ∪ 𝑘 ∈ 𝑦 𝐵 = ∪ 𝑘 ∈ ∅ 𝐵 ) |
3 |
2
|
eleq1d |
⊢ ( 𝑦 = ∅ → ( ∪ 𝑘 ∈ 𝑦 𝐵 ∈ dom vol ↔ ∪ 𝑘 ∈ ∅ 𝐵 ∈ dom vol ) ) |
4 |
1 3
|
imbi12d |
⊢ ( 𝑦 = ∅ → ( ( ∀ 𝑘 ∈ 𝑦 𝐵 ∈ dom vol → ∪ 𝑘 ∈ 𝑦 𝐵 ∈ dom vol ) ↔ ( ∀ 𝑘 ∈ ∅ 𝐵 ∈ dom vol → ∪ 𝑘 ∈ ∅ 𝐵 ∈ dom vol ) ) ) |
5 |
|
raleq |
⊢ ( 𝑦 = 𝑥 → ( ∀ 𝑘 ∈ 𝑦 𝐵 ∈ dom vol ↔ ∀ 𝑘 ∈ 𝑥 𝐵 ∈ dom vol ) ) |
6 |
|
iuneq1 |
⊢ ( 𝑦 = 𝑥 → ∪ 𝑘 ∈ 𝑦 𝐵 = ∪ 𝑘 ∈ 𝑥 𝐵 ) |
7 |
6
|
eleq1d |
⊢ ( 𝑦 = 𝑥 → ( ∪ 𝑘 ∈ 𝑦 𝐵 ∈ dom vol ↔ ∪ 𝑘 ∈ 𝑥 𝐵 ∈ dom vol ) ) |
8 |
5 7
|
imbi12d |
⊢ ( 𝑦 = 𝑥 → ( ( ∀ 𝑘 ∈ 𝑦 𝐵 ∈ dom vol → ∪ 𝑘 ∈ 𝑦 𝐵 ∈ dom vol ) ↔ ( ∀ 𝑘 ∈ 𝑥 𝐵 ∈ dom vol → ∪ 𝑘 ∈ 𝑥 𝐵 ∈ dom vol ) ) ) |
9 |
|
raleq |
⊢ ( 𝑦 = ( 𝑥 ∪ { 𝑧 } ) → ( ∀ 𝑘 ∈ 𝑦 𝐵 ∈ dom vol ↔ ∀ 𝑘 ∈ ( 𝑥 ∪ { 𝑧 } ) 𝐵 ∈ dom vol ) ) |
10 |
|
iuneq1 |
⊢ ( 𝑦 = ( 𝑥 ∪ { 𝑧 } ) → ∪ 𝑘 ∈ 𝑦 𝐵 = ∪ 𝑘 ∈ ( 𝑥 ∪ { 𝑧 } ) 𝐵 ) |
11 |
10
|
eleq1d |
⊢ ( 𝑦 = ( 𝑥 ∪ { 𝑧 } ) → ( ∪ 𝑘 ∈ 𝑦 𝐵 ∈ dom vol ↔ ∪ 𝑘 ∈ ( 𝑥 ∪ { 𝑧 } ) 𝐵 ∈ dom vol ) ) |
12 |
9 11
|
imbi12d |
⊢ ( 𝑦 = ( 𝑥 ∪ { 𝑧 } ) → ( ( ∀ 𝑘 ∈ 𝑦 𝐵 ∈ dom vol → ∪ 𝑘 ∈ 𝑦 𝐵 ∈ dom vol ) ↔ ( ∀ 𝑘 ∈ ( 𝑥 ∪ { 𝑧 } ) 𝐵 ∈ dom vol → ∪ 𝑘 ∈ ( 𝑥 ∪ { 𝑧 } ) 𝐵 ∈ dom vol ) ) ) |
13 |
|
raleq |
⊢ ( 𝑦 = 𝐴 → ( ∀ 𝑘 ∈ 𝑦 𝐵 ∈ dom vol ↔ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ dom vol ) ) |
14 |
|
iuneq1 |
⊢ ( 𝑦 = 𝐴 → ∪ 𝑘 ∈ 𝑦 𝐵 = ∪ 𝑘 ∈ 𝐴 𝐵 ) |
15 |
14
|
eleq1d |
⊢ ( 𝑦 = 𝐴 → ( ∪ 𝑘 ∈ 𝑦 𝐵 ∈ dom vol ↔ ∪ 𝑘 ∈ 𝐴 𝐵 ∈ dom vol ) ) |
16 |
13 15
|
imbi12d |
⊢ ( 𝑦 = 𝐴 → ( ( ∀ 𝑘 ∈ 𝑦 𝐵 ∈ dom vol → ∪ 𝑘 ∈ 𝑦 𝐵 ∈ dom vol ) ↔ ( ∀ 𝑘 ∈ 𝐴 𝐵 ∈ dom vol → ∪ 𝑘 ∈ 𝐴 𝐵 ∈ dom vol ) ) ) |
17 |
|
0iun |
⊢ ∪ 𝑘 ∈ ∅ 𝐵 = ∅ |
18 |
|
0mbl |
⊢ ∅ ∈ dom vol |
19 |
17 18
|
eqeltri |
⊢ ∪ 𝑘 ∈ ∅ 𝐵 ∈ dom vol |
20 |
19
|
a1i |
⊢ ( ∀ 𝑘 ∈ ∅ 𝐵 ∈ dom vol → ∪ 𝑘 ∈ ∅ 𝐵 ∈ dom vol ) |
21 |
|
ssun1 |
⊢ 𝑥 ⊆ ( 𝑥 ∪ { 𝑧 } ) |
22 |
|
ssralv |
⊢ ( 𝑥 ⊆ ( 𝑥 ∪ { 𝑧 } ) → ( ∀ 𝑘 ∈ ( 𝑥 ∪ { 𝑧 } ) 𝐵 ∈ dom vol → ∀ 𝑘 ∈ 𝑥 𝐵 ∈ dom vol ) ) |
23 |
21 22
|
ax-mp |
⊢ ( ∀ 𝑘 ∈ ( 𝑥 ∪ { 𝑧 } ) 𝐵 ∈ dom vol → ∀ 𝑘 ∈ 𝑥 𝐵 ∈ dom vol ) |
24 |
23
|
imim1i |
⊢ ( ( ∀ 𝑘 ∈ 𝑥 𝐵 ∈ dom vol → ∪ 𝑘 ∈ 𝑥 𝐵 ∈ dom vol ) → ( ∀ 𝑘 ∈ ( 𝑥 ∪ { 𝑧 } ) 𝐵 ∈ dom vol → ∪ 𝑘 ∈ 𝑥 𝐵 ∈ dom vol ) ) |
25 |
|
ssun2 |
⊢ { 𝑧 } ⊆ ( 𝑥 ∪ { 𝑧 } ) |
26 |
|
ssralv |
⊢ ( { 𝑧 } ⊆ ( 𝑥 ∪ { 𝑧 } ) → ( ∀ 𝑘 ∈ ( 𝑥 ∪ { 𝑧 } ) 𝐵 ∈ dom vol → ∀ 𝑘 ∈ { 𝑧 } 𝐵 ∈ dom vol ) ) |
27 |
25 26
|
ax-mp |
⊢ ( ∀ 𝑘 ∈ ( 𝑥 ∪ { 𝑧 } ) 𝐵 ∈ dom vol → ∀ 𝑘 ∈ { 𝑧 } 𝐵 ∈ dom vol ) |
28 |
|
iunxun |
⊢ ∪ 𝑘 ∈ ( 𝑥 ∪ { 𝑧 } ) 𝐵 = ( ∪ 𝑘 ∈ 𝑥 𝐵 ∪ ∪ 𝑘 ∈ { 𝑧 } 𝐵 ) |
29 |
|
vex |
⊢ 𝑧 ∈ V |
30 |
|
csbeq1 |
⊢ ( 𝑥 = 𝑧 → ⦋ 𝑥 / 𝑘 ⦌ 𝐵 = ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ) |
31 |
30
|
eleq1d |
⊢ ( 𝑥 = 𝑧 → ( ⦋ 𝑥 / 𝑘 ⦌ 𝐵 ∈ dom vol ↔ ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ∈ dom vol ) ) |
32 |
29 31
|
ralsn |
⊢ ( ∀ 𝑥 ∈ { 𝑧 } ⦋ 𝑥 / 𝑘 ⦌ 𝐵 ∈ dom vol ↔ ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ∈ dom vol ) |
33 |
|
nfv |
⊢ Ⅎ 𝑥 𝐵 ∈ dom vol |
34 |
|
nfcsb1v |
⊢ Ⅎ 𝑘 ⦋ 𝑥 / 𝑘 ⦌ 𝐵 |
35 |
34
|
nfel1 |
⊢ Ⅎ 𝑘 ⦋ 𝑥 / 𝑘 ⦌ 𝐵 ∈ dom vol |
36 |
|
csbeq1a |
⊢ ( 𝑘 = 𝑥 → 𝐵 = ⦋ 𝑥 / 𝑘 ⦌ 𝐵 ) |
37 |
36
|
eleq1d |
⊢ ( 𝑘 = 𝑥 → ( 𝐵 ∈ dom vol ↔ ⦋ 𝑥 / 𝑘 ⦌ 𝐵 ∈ dom vol ) ) |
38 |
33 35 37
|
cbvralw |
⊢ ( ∀ 𝑘 ∈ { 𝑧 } 𝐵 ∈ dom vol ↔ ∀ 𝑥 ∈ { 𝑧 } ⦋ 𝑥 / 𝑘 ⦌ 𝐵 ∈ dom vol ) |
39 |
|
nfcv |
⊢ Ⅎ 𝑥 𝐵 |
40 |
39 34 36
|
cbviun |
⊢ ∪ 𝑘 ∈ { 𝑧 } 𝐵 = ∪ 𝑥 ∈ { 𝑧 } ⦋ 𝑥 / 𝑘 ⦌ 𝐵 |
41 |
29 30
|
iunxsn |
⊢ ∪ 𝑥 ∈ { 𝑧 } ⦋ 𝑥 / 𝑘 ⦌ 𝐵 = ⦋ 𝑧 / 𝑘 ⦌ 𝐵 |
42 |
40 41
|
eqtri |
⊢ ∪ 𝑘 ∈ { 𝑧 } 𝐵 = ⦋ 𝑧 / 𝑘 ⦌ 𝐵 |
43 |
42
|
eleq1i |
⊢ ( ∪ 𝑘 ∈ { 𝑧 } 𝐵 ∈ dom vol ↔ ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ∈ dom vol ) |
44 |
32 38 43
|
3bitr4i |
⊢ ( ∀ 𝑘 ∈ { 𝑧 } 𝐵 ∈ dom vol ↔ ∪ 𝑘 ∈ { 𝑧 } 𝐵 ∈ dom vol ) |
45 |
|
unmbl |
⊢ ( ( ∪ 𝑘 ∈ 𝑥 𝐵 ∈ dom vol ∧ ∪ 𝑘 ∈ { 𝑧 } 𝐵 ∈ dom vol ) → ( ∪ 𝑘 ∈ 𝑥 𝐵 ∪ ∪ 𝑘 ∈ { 𝑧 } 𝐵 ) ∈ dom vol ) |
46 |
44 45
|
sylan2b |
⊢ ( ( ∪ 𝑘 ∈ 𝑥 𝐵 ∈ dom vol ∧ ∀ 𝑘 ∈ { 𝑧 } 𝐵 ∈ dom vol ) → ( ∪ 𝑘 ∈ 𝑥 𝐵 ∪ ∪ 𝑘 ∈ { 𝑧 } 𝐵 ) ∈ dom vol ) |
47 |
28 46
|
eqeltrid |
⊢ ( ( ∪ 𝑘 ∈ 𝑥 𝐵 ∈ dom vol ∧ ∀ 𝑘 ∈ { 𝑧 } 𝐵 ∈ dom vol ) → ∪ 𝑘 ∈ ( 𝑥 ∪ { 𝑧 } ) 𝐵 ∈ dom vol ) |
48 |
47
|
expcom |
⊢ ( ∀ 𝑘 ∈ { 𝑧 } 𝐵 ∈ dom vol → ( ∪ 𝑘 ∈ 𝑥 𝐵 ∈ dom vol → ∪ 𝑘 ∈ ( 𝑥 ∪ { 𝑧 } ) 𝐵 ∈ dom vol ) ) |
49 |
27 48
|
syl |
⊢ ( ∀ 𝑘 ∈ ( 𝑥 ∪ { 𝑧 } ) 𝐵 ∈ dom vol → ( ∪ 𝑘 ∈ 𝑥 𝐵 ∈ dom vol → ∪ 𝑘 ∈ ( 𝑥 ∪ { 𝑧 } ) 𝐵 ∈ dom vol ) ) |
50 |
24 49
|
sylcom |
⊢ ( ( ∀ 𝑘 ∈ 𝑥 𝐵 ∈ dom vol → ∪ 𝑘 ∈ 𝑥 𝐵 ∈ dom vol ) → ( ∀ 𝑘 ∈ ( 𝑥 ∪ { 𝑧 } ) 𝐵 ∈ dom vol → ∪ 𝑘 ∈ ( 𝑥 ∪ { 𝑧 } ) 𝐵 ∈ dom vol ) ) |
51 |
50
|
a1i |
⊢ ( 𝑥 ∈ Fin → ( ( ∀ 𝑘 ∈ 𝑥 𝐵 ∈ dom vol → ∪ 𝑘 ∈ 𝑥 𝐵 ∈ dom vol ) → ( ∀ 𝑘 ∈ ( 𝑥 ∪ { 𝑧 } ) 𝐵 ∈ dom vol → ∪ 𝑘 ∈ ( 𝑥 ∪ { 𝑧 } ) 𝐵 ∈ dom vol ) ) ) |
52 |
4 8 12 16 20 51
|
findcard2 |
⊢ ( 𝐴 ∈ Fin → ( ∀ 𝑘 ∈ 𝐴 𝐵 ∈ dom vol → ∪ 𝑘 ∈ 𝐴 𝐵 ∈ dom vol ) ) |
53 |
52
|
imp |
⊢ ( ( 𝐴 ∈ Fin ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ dom vol ) → ∪ 𝑘 ∈ 𝐴 𝐵 ∈ dom vol ) |