Step |
Hyp |
Ref |
Expression |
1 |
|
efgval.w |
⊢ 𝑊 = ( I ‘ Word ( 𝐼 × 2o ) ) |
2 |
|
efgval.r |
⊢ ∼ = ( ~FG ‘ 𝐼 ) |
3 |
|
efgval2.m |
⊢ 𝑀 = ( 𝑦 ∈ 𝐼 , 𝑧 ∈ 2o ↦ 〈 𝑦 , ( 1o ∖ 𝑧 ) 〉 ) |
4 |
|
efgval2.t |
⊢ 𝑇 = ( 𝑣 ∈ 𝑊 ↦ ( 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) , 𝑤 ∈ ( 𝐼 × 2o ) ↦ ( 𝑣 splice 〈 𝑛 , 𝑛 , 〈“ 𝑤 ( 𝑀 ‘ 𝑤 ) ”〉 〉 ) ) ) |
5 |
|
efgred.d |
⊢ 𝐷 = ( 𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran ( 𝑇 ‘ 𝑥 ) ) |
6 |
|
efgred.s |
⊢ 𝑆 = ( 𝑚 ∈ { 𝑡 ∈ ( Word 𝑊 ∖ { ∅ } ) ∣ ( ( 𝑡 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝑡 ) ) ( 𝑡 ‘ 𝑘 ) ∈ ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑘 − 1 ) ) ) ) } ↦ ( 𝑚 ‘ ( ( ♯ ‘ 𝑚 ) − 1 ) ) ) |
7 |
1 2 3 4 5 6
|
efgsfo |
⊢ 𝑆 : dom 𝑆 –onto→ 𝑊 |
8 |
|
foelrn |
⊢ ( ( 𝑆 : dom 𝑆 –onto→ 𝑊 ∧ 𝐴 ∈ 𝑊 ) → ∃ 𝑎 ∈ dom 𝑆 𝐴 = ( 𝑆 ‘ 𝑎 ) ) |
9 |
7 8
|
mpan |
⊢ ( 𝐴 ∈ 𝑊 → ∃ 𝑎 ∈ dom 𝑆 𝐴 = ( 𝑆 ‘ 𝑎 ) ) |
10 |
1 2 3 4 5 6
|
efgsdm |
⊢ ( 𝑎 ∈ dom 𝑆 ↔ ( 𝑎 ∈ ( Word 𝑊 ∖ { ∅ } ) ∧ ( 𝑎 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑖 ∈ ( 1 ..^ ( ♯ ‘ 𝑎 ) ) ( 𝑎 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝑎 ‘ ( 𝑖 − 1 ) ) ) ) ) |
11 |
10
|
simp2bi |
⊢ ( 𝑎 ∈ dom 𝑆 → ( 𝑎 ‘ 0 ) ∈ 𝐷 ) |
12 |
1 2 3 4 5 6
|
efgsrel |
⊢ ( 𝑎 ∈ dom 𝑆 → ( 𝑎 ‘ 0 ) ∼ ( 𝑆 ‘ 𝑎 ) ) |
13 |
12
|
adantl |
⊢ ( ( 𝐴 ∈ 𝑊 ∧ 𝑎 ∈ dom 𝑆 ) → ( 𝑎 ‘ 0 ) ∼ ( 𝑆 ‘ 𝑎 ) ) |
14 |
|
breq1 |
⊢ ( 𝑑 = ( 𝑎 ‘ 0 ) → ( 𝑑 ∼ ( 𝑆 ‘ 𝑎 ) ↔ ( 𝑎 ‘ 0 ) ∼ ( 𝑆 ‘ 𝑎 ) ) ) |
15 |
14
|
rspcev |
⊢ ( ( ( 𝑎 ‘ 0 ) ∈ 𝐷 ∧ ( 𝑎 ‘ 0 ) ∼ ( 𝑆 ‘ 𝑎 ) ) → ∃ 𝑑 ∈ 𝐷 𝑑 ∼ ( 𝑆 ‘ 𝑎 ) ) |
16 |
11 13 15
|
syl2an2 |
⊢ ( ( 𝐴 ∈ 𝑊 ∧ 𝑎 ∈ dom 𝑆 ) → ∃ 𝑑 ∈ 𝐷 𝑑 ∼ ( 𝑆 ‘ 𝑎 ) ) |
17 |
|
breq2 |
⊢ ( 𝐴 = ( 𝑆 ‘ 𝑎 ) → ( 𝑑 ∼ 𝐴 ↔ 𝑑 ∼ ( 𝑆 ‘ 𝑎 ) ) ) |
18 |
17
|
rexbidv |
⊢ ( 𝐴 = ( 𝑆 ‘ 𝑎 ) → ( ∃ 𝑑 ∈ 𝐷 𝑑 ∼ 𝐴 ↔ ∃ 𝑑 ∈ 𝐷 𝑑 ∼ ( 𝑆 ‘ 𝑎 ) ) ) |
19 |
16 18
|
syl5ibrcom |
⊢ ( ( 𝐴 ∈ 𝑊 ∧ 𝑎 ∈ dom 𝑆 ) → ( 𝐴 = ( 𝑆 ‘ 𝑎 ) → ∃ 𝑑 ∈ 𝐷 𝑑 ∼ 𝐴 ) ) |
20 |
19
|
rexlimdva |
⊢ ( 𝐴 ∈ 𝑊 → ( ∃ 𝑎 ∈ dom 𝑆 𝐴 = ( 𝑆 ‘ 𝑎 ) → ∃ 𝑑 ∈ 𝐷 𝑑 ∼ 𝐴 ) ) |
21 |
9 20
|
mpd |
⊢ ( 𝐴 ∈ 𝑊 → ∃ 𝑑 ∈ 𝐷 𝑑 ∼ 𝐴 ) |
22 |
1 2
|
efger |
⊢ ∼ Er 𝑊 |
23 |
22
|
a1i |
⊢ ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) → ∼ Er 𝑊 ) |
24 |
|
simprl |
⊢ ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) → 𝑑 ∼ 𝐴 ) |
25 |
|
simprr |
⊢ ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) → 𝑐 ∼ 𝐴 ) |
26 |
23 24 25
|
ertr4d |
⊢ ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) → 𝑑 ∼ 𝑐 ) |
27 |
1 2 3 4 5 6
|
efgrelex |
⊢ ( 𝑑 ∼ 𝑐 → ∃ 𝑎 ∈ ( ◡ 𝑆 “ { 𝑑 } ) ∃ 𝑏 ∈ ( ◡ 𝑆 “ { 𝑐 } ) ( 𝑎 ‘ 0 ) = ( 𝑏 ‘ 0 ) ) |
28 |
|
fofn |
⊢ ( 𝑆 : dom 𝑆 –onto→ 𝑊 → 𝑆 Fn dom 𝑆 ) |
29 |
|
fniniseg |
⊢ ( 𝑆 Fn dom 𝑆 → ( 𝑎 ∈ ( ◡ 𝑆 “ { 𝑑 } ) ↔ ( 𝑎 ∈ dom 𝑆 ∧ ( 𝑆 ‘ 𝑎 ) = 𝑑 ) ) ) |
30 |
7 28 29
|
mp2b |
⊢ ( 𝑎 ∈ ( ◡ 𝑆 “ { 𝑑 } ) ↔ ( 𝑎 ∈ dom 𝑆 ∧ ( 𝑆 ‘ 𝑎 ) = 𝑑 ) ) |
31 |
30
|
simplbi |
⊢ ( 𝑎 ∈ ( ◡ 𝑆 “ { 𝑑 } ) → 𝑎 ∈ dom 𝑆 ) |
32 |
31
|
ad2antrl |
⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ◡ 𝑆 “ { 𝑐 } ) ) ) → 𝑎 ∈ dom 𝑆 ) |
33 |
1 2 3 4 5 6
|
efgsval |
⊢ ( 𝑎 ∈ dom 𝑆 → ( 𝑆 ‘ 𝑎 ) = ( 𝑎 ‘ ( ( ♯ ‘ 𝑎 ) − 1 ) ) ) |
34 |
32 33
|
syl |
⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ◡ 𝑆 “ { 𝑐 } ) ) ) → ( 𝑆 ‘ 𝑎 ) = ( 𝑎 ‘ ( ( ♯ ‘ 𝑎 ) − 1 ) ) ) |
35 |
30
|
simprbi |
⊢ ( 𝑎 ∈ ( ◡ 𝑆 “ { 𝑑 } ) → ( 𝑆 ‘ 𝑎 ) = 𝑑 ) |
36 |
35
|
ad2antrl |
⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ◡ 𝑆 “ { 𝑐 } ) ) ) → ( 𝑆 ‘ 𝑎 ) = 𝑑 ) |
37 |
|
simpllr |
⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ◡ 𝑆 “ { 𝑐 } ) ) ) → ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) |
38 |
37
|
simpld |
⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ◡ 𝑆 “ { 𝑐 } ) ) ) → 𝑑 ∈ 𝐷 ) |
39 |
36 38
|
eqeltrd |
⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ◡ 𝑆 “ { 𝑐 } ) ) ) → ( 𝑆 ‘ 𝑎 ) ∈ 𝐷 ) |
40 |
1 2 3 4 5 6
|
efgs1b |
⊢ ( 𝑎 ∈ dom 𝑆 → ( ( 𝑆 ‘ 𝑎 ) ∈ 𝐷 ↔ ( ♯ ‘ 𝑎 ) = 1 ) ) |
41 |
32 40
|
syl |
⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ◡ 𝑆 “ { 𝑐 } ) ) ) → ( ( 𝑆 ‘ 𝑎 ) ∈ 𝐷 ↔ ( ♯ ‘ 𝑎 ) = 1 ) ) |
42 |
39 41
|
mpbid |
⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ◡ 𝑆 “ { 𝑐 } ) ) ) → ( ♯ ‘ 𝑎 ) = 1 ) |
43 |
42
|
oveq1d |
⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ◡ 𝑆 “ { 𝑐 } ) ) ) → ( ( ♯ ‘ 𝑎 ) − 1 ) = ( 1 − 1 ) ) |
44 |
|
1m1e0 |
⊢ ( 1 − 1 ) = 0 |
45 |
43 44
|
eqtrdi |
⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ◡ 𝑆 “ { 𝑐 } ) ) ) → ( ( ♯ ‘ 𝑎 ) − 1 ) = 0 ) |
46 |
45
|
fveq2d |
⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ◡ 𝑆 “ { 𝑐 } ) ) ) → ( 𝑎 ‘ ( ( ♯ ‘ 𝑎 ) − 1 ) ) = ( 𝑎 ‘ 0 ) ) |
47 |
34 36 46
|
3eqtr3rd |
⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ◡ 𝑆 “ { 𝑐 } ) ) ) → ( 𝑎 ‘ 0 ) = 𝑑 ) |
48 |
|
fniniseg |
⊢ ( 𝑆 Fn dom 𝑆 → ( 𝑏 ∈ ( ◡ 𝑆 “ { 𝑐 } ) ↔ ( 𝑏 ∈ dom 𝑆 ∧ ( 𝑆 ‘ 𝑏 ) = 𝑐 ) ) ) |
49 |
7 28 48
|
mp2b |
⊢ ( 𝑏 ∈ ( ◡ 𝑆 “ { 𝑐 } ) ↔ ( 𝑏 ∈ dom 𝑆 ∧ ( 𝑆 ‘ 𝑏 ) = 𝑐 ) ) |
50 |
49
|
simplbi |
⊢ ( 𝑏 ∈ ( ◡ 𝑆 “ { 𝑐 } ) → 𝑏 ∈ dom 𝑆 ) |
51 |
50
|
ad2antll |
⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ◡ 𝑆 “ { 𝑐 } ) ) ) → 𝑏 ∈ dom 𝑆 ) |
52 |
1 2 3 4 5 6
|
efgsval |
⊢ ( 𝑏 ∈ dom 𝑆 → ( 𝑆 ‘ 𝑏 ) = ( 𝑏 ‘ ( ( ♯ ‘ 𝑏 ) − 1 ) ) ) |
53 |
51 52
|
syl |
⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ◡ 𝑆 “ { 𝑐 } ) ) ) → ( 𝑆 ‘ 𝑏 ) = ( 𝑏 ‘ ( ( ♯ ‘ 𝑏 ) − 1 ) ) ) |
54 |
49
|
simprbi |
⊢ ( 𝑏 ∈ ( ◡ 𝑆 “ { 𝑐 } ) → ( 𝑆 ‘ 𝑏 ) = 𝑐 ) |
55 |
54
|
ad2antll |
⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ◡ 𝑆 “ { 𝑐 } ) ) ) → ( 𝑆 ‘ 𝑏 ) = 𝑐 ) |
56 |
37
|
simprd |
⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ◡ 𝑆 “ { 𝑐 } ) ) ) → 𝑐 ∈ 𝐷 ) |
57 |
55 56
|
eqeltrd |
⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ◡ 𝑆 “ { 𝑐 } ) ) ) → ( 𝑆 ‘ 𝑏 ) ∈ 𝐷 ) |
58 |
1 2 3 4 5 6
|
efgs1b |
⊢ ( 𝑏 ∈ dom 𝑆 → ( ( 𝑆 ‘ 𝑏 ) ∈ 𝐷 ↔ ( ♯ ‘ 𝑏 ) = 1 ) ) |
59 |
51 58
|
syl |
⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ◡ 𝑆 “ { 𝑐 } ) ) ) → ( ( 𝑆 ‘ 𝑏 ) ∈ 𝐷 ↔ ( ♯ ‘ 𝑏 ) = 1 ) ) |
60 |
57 59
|
mpbid |
⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ◡ 𝑆 “ { 𝑐 } ) ) ) → ( ♯ ‘ 𝑏 ) = 1 ) |
61 |
60
|
oveq1d |
⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ◡ 𝑆 “ { 𝑐 } ) ) ) → ( ( ♯ ‘ 𝑏 ) − 1 ) = ( 1 − 1 ) ) |
62 |
61 44
|
eqtrdi |
⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ◡ 𝑆 “ { 𝑐 } ) ) ) → ( ( ♯ ‘ 𝑏 ) − 1 ) = 0 ) |
63 |
62
|
fveq2d |
⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ◡ 𝑆 “ { 𝑐 } ) ) ) → ( 𝑏 ‘ ( ( ♯ ‘ 𝑏 ) − 1 ) ) = ( 𝑏 ‘ 0 ) ) |
64 |
53 55 63
|
3eqtr3rd |
⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ◡ 𝑆 “ { 𝑐 } ) ) ) → ( 𝑏 ‘ 0 ) = 𝑐 ) |
65 |
47 64
|
eqeq12d |
⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ◡ 𝑆 “ { 𝑐 } ) ) ) → ( ( 𝑎 ‘ 0 ) = ( 𝑏 ‘ 0 ) ↔ 𝑑 = 𝑐 ) ) |
66 |
65
|
biimpd |
⊢ ( ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) ∧ ( 𝑎 ∈ ( ◡ 𝑆 “ { 𝑑 } ) ∧ 𝑏 ∈ ( ◡ 𝑆 “ { 𝑐 } ) ) ) → ( ( 𝑎 ‘ 0 ) = ( 𝑏 ‘ 0 ) → 𝑑 = 𝑐 ) ) |
67 |
66
|
rexlimdvva |
⊢ ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) → ( ∃ 𝑎 ∈ ( ◡ 𝑆 “ { 𝑑 } ) ∃ 𝑏 ∈ ( ◡ 𝑆 “ { 𝑐 } ) ( 𝑎 ‘ 0 ) = ( 𝑏 ‘ 0 ) → 𝑑 = 𝑐 ) ) |
68 |
27 67
|
syl5 |
⊢ ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) → ( 𝑑 ∼ 𝑐 → 𝑑 = 𝑐 ) ) |
69 |
26 68
|
mpd |
⊢ ( ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) ∧ ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) ) → 𝑑 = 𝑐 ) |
70 |
69
|
ex |
⊢ ( ( 𝐴 ∈ 𝑊 ∧ ( 𝑑 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷 ) ) → ( ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) → 𝑑 = 𝑐 ) ) |
71 |
70
|
ralrimivva |
⊢ ( 𝐴 ∈ 𝑊 → ∀ 𝑑 ∈ 𝐷 ∀ 𝑐 ∈ 𝐷 ( ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) → 𝑑 = 𝑐 ) ) |
72 |
|
breq1 |
⊢ ( 𝑑 = 𝑐 → ( 𝑑 ∼ 𝐴 ↔ 𝑐 ∼ 𝐴 ) ) |
73 |
72
|
reu4 |
⊢ ( ∃! 𝑑 ∈ 𝐷 𝑑 ∼ 𝐴 ↔ ( ∃ 𝑑 ∈ 𝐷 𝑑 ∼ 𝐴 ∧ ∀ 𝑑 ∈ 𝐷 ∀ 𝑐 ∈ 𝐷 ( ( 𝑑 ∼ 𝐴 ∧ 𝑐 ∼ 𝐴 ) → 𝑑 = 𝑐 ) ) ) |
74 |
21 71 73
|
sylanbrc |
⊢ ( 𝐴 ∈ 𝑊 → ∃! 𝑑 ∈ 𝐷 𝑑 ∼ 𝐴 ) |