| Step |
Hyp |
Ref |
Expression |
| 1 |
|
symgtrf.t |
⊢ 𝑇 = ran ( pmTrsp ‘ 𝐷 ) |
| 2 |
|
symgtrf.g |
⊢ 𝐺 = ( SymGrp ‘ 𝐷 ) |
| 3 |
|
symgtrf.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
| 4 |
|
symggen.k |
⊢ 𝐾 = ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) |
| 5 |
|
elex |
⊢ ( 𝐷 ∈ 𝑉 → 𝐷 ∈ V ) |
| 6 |
2
|
symggrp |
⊢ ( 𝐷 ∈ V → 𝐺 ∈ Grp ) |
| 7 |
|
grpmnd |
⊢ ( 𝐺 ∈ Grp → 𝐺 ∈ Mnd ) |
| 8 |
6 7
|
syl |
⊢ ( 𝐷 ∈ V → 𝐺 ∈ Mnd ) |
| 9 |
3
|
submacs |
⊢ ( 𝐺 ∈ Mnd → ( SubMnd ‘ 𝐺 ) ∈ ( ACS ‘ 𝐵 ) ) |
| 10 |
|
acsmre |
⊢ ( ( SubMnd ‘ 𝐺 ) ∈ ( ACS ‘ 𝐵 ) → ( SubMnd ‘ 𝐺 ) ∈ ( Moore ‘ 𝐵 ) ) |
| 11 |
8 9 10
|
3syl |
⊢ ( 𝐷 ∈ V → ( SubMnd ‘ 𝐺 ) ∈ ( Moore ‘ 𝐵 ) ) |
| 12 |
5 11
|
syl |
⊢ ( 𝐷 ∈ 𝑉 → ( SubMnd ‘ 𝐺 ) ∈ ( Moore ‘ 𝐵 ) ) |
| 13 |
1 2 3
|
symgtrf |
⊢ 𝑇 ⊆ 𝐵 |
| 14 |
13
|
a1i |
⊢ ( 𝐷 ∈ 𝑉 → 𝑇 ⊆ 𝐵 ) |
| 15 |
|
2onn |
⊢ 2o ∈ ω |
| 16 |
|
nnfi |
⊢ ( 2o ∈ ω → 2o ∈ Fin ) |
| 17 |
15 16
|
ax-mp |
⊢ 2o ∈ Fin |
| 18 |
|
eqid |
⊢ ( pmTrsp ‘ 𝐷 ) = ( pmTrsp ‘ 𝐷 ) |
| 19 |
18 1
|
pmtrfb |
⊢ ( 𝑥 ∈ 𝑇 ↔ ( 𝐷 ∈ V ∧ 𝑥 : 𝐷 –1-1-onto→ 𝐷 ∧ dom ( 𝑥 ∖ I ) ≈ 2o ) ) |
| 20 |
19
|
simp3bi |
⊢ ( 𝑥 ∈ 𝑇 → dom ( 𝑥 ∖ I ) ≈ 2o ) |
| 21 |
|
enfi |
⊢ ( dom ( 𝑥 ∖ I ) ≈ 2o → ( dom ( 𝑥 ∖ I ) ∈ Fin ↔ 2o ∈ Fin ) ) |
| 22 |
20 21
|
syl |
⊢ ( 𝑥 ∈ 𝑇 → ( dom ( 𝑥 ∖ I ) ∈ Fin ↔ 2o ∈ Fin ) ) |
| 23 |
22
|
adantl |
⊢ ( ( 𝐷 ∈ 𝑉 ∧ 𝑥 ∈ 𝑇 ) → ( dom ( 𝑥 ∖ I ) ∈ Fin ↔ 2o ∈ Fin ) ) |
| 24 |
17 23
|
mpbiri |
⊢ ( ( 𝐷 ∈ 𝑉 ∧ 𝑥 ∈ 𝑇 ) → dom ( 𝑥 ∖ I ) ∈ Fin ) |
| 25 |
14 24
|
ssrabdv |
⊢ ( 𝐷 ∈ 𝑉 → 𝑇 ⊆ { 𝑥 ∈ 𝐵 ∣ dom ( 𝑥 ∖ I ) ∈ Fin } ) |
| 26 |
2 3
|
symgfisg |
⊢ ( 𝐷 ∈ 𝑉 → { 𝑥 ∈ 𝐵 ∣ dom ( 𝑥 ∖ I ) ∈ Fin } ∈ ( SubGrp ‘ 𝐺 ) ) |
| 27 |
|
subgsubm |
⊢ ( { 𝑥 ∈ 𝐵 ∣ dom ( 𝑥 ∖ I ) ∈ Fin } ∈ ( SubGrp ‘ 𝐺 ) → { 𝑥 ∈ 𝐵 ∣ dom ( 𝑥 ∖ I ) ∈ Fin } ∈ ( SubMnd ‘ 𝐺 ) ) |
| 28 |
26 27
|
syl |
⊢ ( 𝐷 ∈ 𝑉 → { 𝑥 ∈ 𝐵 ∣ dom ( 𝑥 ∖ I ) ∈ Fin } ∈ ( SubMnd ‘ 𝐺 ) ) |
| 29 |
4
|
mrcsscl |
⊢ ( ( ( SubMnd ‘ 𝐺 ) ∈ ( Moore ‘ 𝐵 ) ∧ 𝑇 ⊆ { 𝑥 ∈ 𝐵 ∣ dom ( 𝑥 ∖ I ) ∈ Fin } ∧ { 𝑥 ∈ 𝐵 ∣ dom ( 𝑥 ∖ I ) ∈ Fin } ∈ ( SubMnd ‘ 𝐺 ) ) → ( 𝐾 ‘ 𝑇 ) ⊆ { 𝑥 ∈ 𝐵 ∣ dom ( 𝑥 ∖ I ) ∈ Fin } ) |
| 30 |
12 25 28 29
|
syl3anc |
⊢ ( 𝐷 ∈ 𝑉 → ( 𝐾 ‘ 𝑇 ) ⊆ { 𝑥 ∈ 𝐵 ∣ dom ( 𝑥 ∖ I ) ∈ Fin } ) |
| 31 |
|
vex |
⊢ 𝑥 ∈ V |
| 32 |
31
|
a1i |
⊢ ( dom ( 𝑥 ∖ I ) ∈ Fin → 𝑥 ∈ V ) |
| 33 |
|
finnum |
⊢ ( dom ( 𝑥 ∖ I ) ∈ Fin → dom ( 𝑥 ∖ I ) ∈ dom card ) |
| 34 |
|
domfi |
⊢ ( ( dom ( 𝑥 ∖ I ) ∈ Fin ∧ dom ( 𝑦 ∖ I ) ≼ dom ( 𝑥 ∖ I ) ) → dom ( 𝑦 ∖ I ) ∈ Fin ) |
| 35 |
2 3
|
symgbasf1o |
⊢ ( 𝑦 ∈ 𝐵 → 𝑦 : 𝐷 –1-1-onto→ 𝐷 ) |
| 36 |
35
|
adantl |
⊢ ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) → 𝑦 : 𝐷 –1-1-onto→ 𝐷 ) |
| 37 |
|
f1ofn |
⊢ ( 𝑦 : 𝐷 –1-1-onto→ 𝐷 → 𝑦 Fn 𝐷 ) |
| 38 |
|
fnnfpeq0 |
⊢ ( 𝑦 Fn 𝐷 → ( dom ( 𝑦 ∖ I ) = ∅ ↔ 𝑦 = ( I ↾ 𝐷 ) ) ) |
| 39 |
36 37 38
|
3syl |
⊢ ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) → ( dom ( 𝑦 ∖ I ) = ∅ ↔ 𝑦 = ( I ↾ 𝐷 ) ) ) |
| 40 |
2 3
|
elbasfv |
⊢ ( 𝑦 ∈ 𝐵 → 𝐷 ∈ V ) |
| 41 |
40
|
adantl |
⊢ ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) → 𝐷 ∈ V ) |
| 42 |
2
|
symgid |
⊢ ( 𝐷 ∈ V → ( I ↾ 𝐷 ) = ( 0g ‘ 𝐺 ) ) |
| 43 |
41 42
|
syl |
⊢ ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) → ( I ↾ 𝐷 ) = ( 0g ‘ 𝐺 ) ) |
| 44 |
41 11
|
syl |
⊢ ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) → ( SubMnd ‘ 𝐺 ) ∈ ( Moore ‘ 𝐵 ) ) |
| 45 |
4
|
mrccl |
⊢ ( ( ( SubMnd ‘ 𝐺 ) ∈ ( Moore ‘ 𝐵 ) ∧ 𝑇 ⊆ 𝐵 ) → ( 𝐾 ‘ 𝑇 ) ∈ ( SubMnd ‘ 𝐺 ) ) |
| 46 |
44 13 45
|
sylancl |
⊢ ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) → ( 𝐾 ‘ 𝑇 ) ∈ ( SubMnd ‘ 𝐺 ) ) |
| 47 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
| 48 |
47
|
subm0cl |
⊢ ( ( 𝐾 ‘ 𝑇 ) ∈ ( SubMnd ‘ 𝐺 ) → ( 0g ‘ 𝐺 ) ∈ ( 𝐾 ‘ 𝑇 ) ) |
| 49 |
46 48
|
syl |
⊢ ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) → ( 0g ‘ 𝐺 ) ∈ ( 𝐾 ‘ 𝑇 ) ) |
| 50 |
43 49
|
eqeltrd |
⊢ ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) → ( I ↾ 𝐷 ) ∈ ( 𝐾 ‘ 𝑇 ) ) |
| 51 |
|
eleq1a |
⊢ ( ( I ↾ 𝐷 ) ∈ ( 𝐾 ‘ 𝑇 ) → ( 𝑦 = ( I ↾ 𝐷 ) → 𝑦 ∈ ( 𝐾 ‘ 𝑇 ) ) ) |
| 52 |
50 51
|
syl |
⊢ ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) → ( 𝑦 = ( I ↾ 𝐷 ) → 𝑦 ∈ ( 𝐾 ‘ 𝑇 ) ) ) |
| 53 |
39 52
|
sylbid |
⊢ ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) → ( dom ( 𝑦 ∖ I ) = ∅ → 𝑦 ∈ ( 𝐾 ‘ 𝑇 ) ) ) |
| 54 |
53
|
adantr |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ ∀ 𝑧 ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) → ( dom ( 𝑦 ∖ I ) = ∅ → 𝑦 ∈ ( 𝐾 ‘ 𝑇 ) ) ) |
| 55 |
|
n0 |
⊢ ( dom ( 𝑦 ∖ I ) ≠ ∅ ↔ ∃ 𝑢 𝑢 ∈ dom ( 𝑦 ∖ I ) ) |
| 56 |
41
|
adantr |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → 𝐷 ∈ V ) |
| 57 |
|
simpr |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → 𝑢 ∈ dom ( 𝑦 ∖ I ) ) |
| 58 |
|
f1omvdmvd |
⊢ ( ( 𝑦 : 𝐷 –1-1-onto→ 𝐷 ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( 𝑦 ‘ 𝑢 ) ∈ ( dom ( 𝑦 ∖ I ) ∖ { 𝑢 } ) ) |
| 59 |
36 58
|
sylan |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( 𝑦 ‘ 𝑢 ) ∈ ( dom ( 𝑦 ∖ I ) ∖ { 𝑢 } ) ) |
| 60 |
59
|
eldifad |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( 𝑦 ‘ 𝑢 ) ∈ dom ( 𝑦 ∖ I ) ) |
| 61 |
57 60
|
prssd |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ⊆ dom ( 𝑦 ∖ I ) ) |
| 62 |
|
difss |
⊢ ( 𝑦 ∖ I ) ⊆ 𝑦 |
| 63 |
|
dmss |
⊢ ( ( 𝑦 ∖ I ) ⊆ 𝑦 → dom ( 𝑦 ∖ I ) ⊆ dom 𝑦 ) |
| 64 |
62 63
|
ax-mp |
⊢ dom ( 𝑦 ∖ I ) ⊆ dom 𝑦 |
| 65 |
|
f1odm |
⊢ ( 𝑦 : 𝐷 –1-1-onto→ 𝐷 → dom 𝑦 = 𝐷 ) |
| 66 |
36 65
|
syl |
⊢ ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) → dom 𝑦 = 𝐷 ) |
| 67 |
64 66
|
sseqtrid |
⊢ ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) → dom ( 𝑦 ∖ I ) ⊆ 𝐷 ) |
| 68 |
67
|
adantr |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → dom ( 𝑦 ∖ I ) ⊆ 𝐷 ) |
| 69 |
61 68
|
sstrd |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ⊆ 𝐷 ) |
| 70 |
|
vex |
⊢ 𝑢 ∈ V |
| 71 |
|
fvex |
⊢ ( 𝑦 ‘ 𝑢 ) ∈ V |
| 72 |
36
|
adantr |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → 𝑦 : 𝐷 –1-1-onto→ 𝐷 ) |
| 73 |
72 37
|
syl |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → 𝑦 Fn 𝐷 ) |
| 74 |
67
|
sselda |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → 𝑢 ∈ 𝐷 ) |
| 75 |
|
fnelnfp |
⊢ ( ( 𝑦 Fn 𝐷 ∧ 𝑢 ∈ 𝐷 ) → ( 𝑢 ∈ dom ( 𝑦 ∖ I ) ↔ ( 𝑦 ‘ 𝑢 ) ≠ 𝑢 ) ) |
| 76 |
73 74 75
|
syl2anc |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( 𝑢 ∈ dom ( 𝑦 ∖ I ) ↔ ( 𝑦 ‘ 𝑢 ) ≠ 𝑢 ) ) |
| 77 |
57 76
|
mpbid |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( 𝑦 ‘ 𝑢 ) ≠ 𝑢 ) |
| 78 |
77
|
necomd |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → 𝑢 ≠ ( 𝑦 ‘ 𝑢 ) ) |
| 79 |
|
pr2nelem |
⊢ ( ( 𝑢 ∈ V ∧ ( 𝑦 ‘ 𝑢 ) ∈ V ∧ 𝑢 ≠ ( 𝑦 ‘ 𝑢 ) ) → { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ≈ 2o ) |
| 80 |
70 71 78 79
|
mp3an12i |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ≈ 2o ) |
| 81 |
18 1
|
pmtrrn |
⊢ ( ( 𝐷 ∈ V ∧ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ⊆ 𝐷 ∧ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ≈ 2o ) → ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∈ 𝑇 ) |
| 82 |
56 69 80 81
|
syl3anc |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∈ 𝑇 ) |
| 83 |
13 82
|
sseldi |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∈ 𝐵 ) |
| 84 |
|
simplr |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → 𝑦 ∈ 𝐵 ) |
| 85 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
| 86 |
2 3 85
|
symgov |
⊢ ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) = ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ) |
| 87 |
83 84 86
|
syl2anc |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) = ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ) |
| 88 |
87
|
oveq2d |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ) = ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ) ) |
| 89 |
41 6
|
syl |
⊢ ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) → 𝐺 ∈ Grp ) |
| 90 |
89
|
adantr |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → 𝐺 ∈ Grp ) |
| 91 |
3 85
|
grpcl |
⊢ ( ( 𝐺 ∈ Grp ∧ ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝐵 ) |
| 92 |
90 83 84 91
|
syl3anc |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝐵 ) |
| 93 |
87 92
|
eqeltrrd |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ∈ 𝐵 ) |
| 94 |
2 3 85
|
symgov |
⊢ ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∈ 𝐵 ∧ ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ∈ 𝐵 ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ) = ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ) ) |
| 95 |
83 93 94
|
syl2anc |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ) = ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ) ) |
| 96 |
|
coass |
⊢ ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ) ∘ 𝑦 ) = ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ) |
| 97 |
18 1
|
pmtrfinv |
⊢ ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∈ 𝑇 → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ) = ( I ↾ 𝐷 ) ) |
| 98 |
82 97
|
syl |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ) = ( I ↾ 𝐷 ) ) |
| 99 |
98
|
coeq1d |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ) ∘ 𝑦 ) = ( ( I ↾ 𝐷 ) ∘ 𝑦 ) ) |
| 100 |
|
f1of |
⊢ ( 𝑦 : 𝐷 –1-1-onto→ 𝐷 → 𝑦 : 𝐷 ⟶ 𝐷 ) |
| 101 |
|
fcoi2 |
⊢ ( 𝑦 : 𝐷 ⟶ 𝐷 → ( ( I ↾ 𝐷 ) ∘ 𝑦 ) = 𝑦 ) |
| 102 |
72 100 101
|
3syl |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( I ↾ 𝐷 ) ∘ 𝑦 ) = 𝑦 ) |
| 103 |
99 102
|
eqtrd |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ) ∘ 𝑦 ) = 𝑦 ) |
| 104 |
96 103
|
eqtr3id |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ) = 𝑦 ) |
| 105 |
88 95 104
|
3eqtrd |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ) = 𝑦 ) |
| 106 |
105
|
adantlr |
⊢ ( ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ ∀ 𝑧 ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ) = 𝑦 ) |
| 107 |
46
|
ad2antrr |
⊢ ( ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ ∀ 𝑧 ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( 𝐾 ‘ 𝑇 ) ∈ ( SubMnd ‘ 𝐺 ) ) |
| 108 |
4
|
mrcssid |
⊢ ( ( ( SubMnd ‘ 𝐺 ) ∈ ( Moore ‘ 𝐵 ) ∧ 𝑇 ⊆ 𝐵 ) → 𝑇 ⊆ ( 𝐾 ‘ 𝑇 ) ) |
| 109 |
44 13 108
|
sylancl |
⊢ ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) → 𝑇 ⊆ ( 𝐾 ‘ 𝑇 ) ) |
| 110 |
109
|
adantr |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → 𝑇 ⊆ ( 𝐾 ‘ 𝑇 ) ) |
| 111 |
110 82
|
sseldd |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∈ ( 𝐾 ‘ 𝑇 ) ) |
| 112 |
111
|
adantlr |
⊢ ( ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ ∀ 𝑧 ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∈ ( 𝐾 ‘ 𝑇 ) ) |
| 113 |
87
|
difeq1d |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∖ I ) = ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ∖ I ) ) |
| 114 |
113
|
dmeqd |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → dom ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∖ I ) = dom ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ∖ I ) ) |
| 115 |
|
simpll |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → dom ( 𝑦 ∖ I ) ∈ Fin ) |
| 116 |
|
mvdco |
⊢ dom ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ∖ I ) ⊆ ( dom ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∖ I ) ∪ dom ( 𝑦 ∖ I ) ) |
| 117 |
18
|
pmtrmvd |
⊢ ( ( 𝐷 ∈ V ∧ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ⊆ 𝐷 ∧ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ≈ 2o ) → dom ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∖ I ) = { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) |
| 118 |
56 69 80 117
|
syl3anc |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → dom ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∖ I ) = { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) |
| 119 |
118 61
|
eqsstrd |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → dom ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∖ I ) ⊆ dom ( 𝑦 ∖ I ) ) |
| 120 |
|
ssidd |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → dom ( 𝑦 ∖ I ) ⊆ dom ( 𝑦 ∖ I ) ) |
| 121 |
119 120
|
unssd |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( dom ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∖ I ) ∪ dom ( 𝑦 ∖ I ) ) ⊆ dom ( 𝑦 ∖ I ) ) |
| 122 |
116 121
|
sstrid |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → dom ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ∖ I ) ⊆ dom ( 𝑦 ∖ I ) ) |
| 123 |
|
fvco2 |
⊢ ( ( 𝑦 Fn 𝐷 ∧ 𝑢 ∈ 𝐷 ) → ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ‘ 𝑢 ) = ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ‘ ( 𝑦 ‘ 𝑢 ) ) ) |
| 124 |
73 74 123
|
syl2anc |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ‘ 𝑢 ) = ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ‘ ( 𝑦 ‘ 𝑢 ) ) ) |
| 125 |
|
prcom |
⊢ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } = { ( 𝑦 ‘ 𝑢 ) , 𝑢 } |
| 126 |
125
|
fveq2i |
⊢ ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) = ( ( pmTrsp ‘ 𝐷 ) ‘ { ( 𝑦 ‘ 𝑢 ) , 𝑢 } ) |
| 127 |
126
|
fveq1i |
⊢ ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ‘ ( 𝑦 ‘ 𝑢 ) ) = ( ( ( pmTrsp ‘ 𝐷 ) ‘ { ( 𝑦 ‘ 𝑢 ) , 𝑢 } ) ‘ ( 𝑦 ‘ 𝑢 ) ) |
| 128 |
68 60
|
sseldd |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( 𝑦 ‘ 𝑢 ) ∈ 𝐷 ) |
| 129 |
18
|
pmtrprfv |
⊢ ( ( 𝐷 ∈ V ∧ ( ( 𝑦 ‘ 𝑢 ) ∈ 𝐷 ∧ 𝑢 ∈ 𝐷 ∧ ( 𝑦 ‘ 𝑢 ) ≠ 𝑢 ) ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { ( 𝑦 ‘ 𝑢 ) , 𝑢 } ) ‘ ( 𝑦 ‘ 𝑢 ) ) = 𝑢 ) |
| 130 |
56 128 74 77 129
|
syl13anc |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { ( 𝑦 ‘ 𝑢 ) , 𝑢 } ) ‘ ( 𝑦 ‘ 𝑢 ) ) = 𝑢 ) |
| 131 |
127 130
|
syl5eq |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ‘ ( 𝑦 ‘ 𝑢 ) ) = 𝑢 ) |
| 132 |
124 131
|
eqtrd |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ‘ 𝑢 ) = 𝑢 ) |
| 133 |
2 3
|
symgbasf1o |
⊢ ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ∈ 𝐵 → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) : 𝐷 –1-1-onto→ 𝐷 ) |
| 134 |
|
f1ofn |
⊢ ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) : 𝐷 –1-1-onto→ 𝐷 → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) Fn 𝐷 ) |
| 135 |
93 133 134
|
3syl |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) Fn 𝐷 ) |
| 136 |
|
fnelnfp |
⊢ ( ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) Fn 𝐷 ∧ 𝑢 ∈ 𝐷 ) → ( 𝑢 ∈ dom ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ∖ I ) ↔ ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ‘ 𝑢 ) ≠ 𝑢 ) ) |
| 137 |
136
|
necon2bbid |
⊢ ( ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) Fn 𝐷 ∧ 𝑢 ∈ 𝐷 ) → ( ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ‘ 𝑢 ) = 𝑢 ↔ ¬ 𝑢 ∈ dom ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ∖ I ) ) ) |
| 138 |
135 74 137
|
syl2anc |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ‘ 𝑢 ) = 𝑢 ↔ ¬ 𝑢 ∈ dom ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ∖ I ) ) ) |
| 139 |
132 138
|
mpbid |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ¬ 𝑢 ∈ dom ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ∖ I ) ) |
| 140 |
122 57 139
|
ssnelpssd |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → dom ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ∖ I ) ⊊ dom ( 𝑦 ∖ I ) ) |
| 141 |
|
php3 |
⊢ ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ dom ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ∖ I ) ⊊ dom ( 𝑦 ∖ I ) ) → dom ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ∖ I ) ≺ dom ( 𝑦 ∖ I ) ) |
| 142 |
115 140 141
|
syl2anc |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → dom ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ∖ I ) ≺ dom ( 𝑦 ∖ I ) ) |
| 143 |
114 142
|
eqbrtrd |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → dom ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∖ I ) ≺ dom ( 𝑦 ∖ I ) ) |
| 144 |
143
|
adantlr |
⊢ ( ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ ∀ 𝑧 ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → dom ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∖ I ) ≺ dom ( 𝑦 ∖ I ) ) |
| 145 |
92
|
adantlr |
⊢ ( ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ ∀ 𝑧 ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝐵 ) |
| 146 |
|
ovex |
⊢ ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ V |
| 147 |
|
difeq1 |
⊢ ( 𝑧 = ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) → ( 𝑧 ∖ I ) = ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∖ I ) ) |
| 148 |
147
|
dmeqd |
⊢ ( 𝑧 = ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) → dom ( 𝑧 ∖ I ) = dom ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∖ I ) ) |
| 149 |
148
|
breq1d |
⊢ ( 𝑧 = ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) → ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) ↔ dom ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∖ I ) ≺ dom ( 𝑦 ∖ I ) ) ) |
| 150 |
|
eleq1 |
⊢ ( 𝑧 = ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) → ( 𝑧 ∈ 𝐵 ↔ ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝐵 ) ) |
| 151 |
|
eleq1 |
⊢ ( 𝑧 = ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) → ( 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ↔ ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ ( 𝐾 ‘ 𝑇 ) ) ) |
| 152 |
150 151
|
imbi12d |
⊢ ( 𝑧 = ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) → ( ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ↔ ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝐵 → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) |
| 153 |
149 152
|
imbi12d |
⊢ ( 𝑧 = ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) → ( ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ↔ ( dom ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝐵 → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) ) |
| 154 |
146 153
|
spcv |
⊢ ( ∀ 𝑧 ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) → ( dom ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝐵 → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) |
| 155 |
154
|
ad2antlr |
⊢ ( ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ ∀ 𝑧 ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( dom ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝐵 → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) |
| 156 |
144 145 155
|
mp2d |
⊢ ( ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ ∀ 𝑧 ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ ( 𝐾 ‘ 𝑇 ) ) |
| 157 |
85
|
submcl |
⊢ ( ( ( 𝐾 ‘ 𝑇 ) ∈ ( SubMnd ‘ 𝐺 ) ∧ ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∈ ( 𝐾 ‘ 𝑇 ) ∧ ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ ( 𝐾 ‘ 𝑇 ) ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ) ∈ ( 𝐾 ‘ 𝑇 ) ) |
| 158 |
107 112 156 157
|
syl3anc |
⊢ ( ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ ∀ 𝑧 ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ) ∈ ( 𝐾 ‘ 𝑇 ) ) |
| 159 |
106 158
|
eqeltrrd |
⊢ ( ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ ∀ 𝑧 ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → 𝑦 ∈ ( 𝐾 ‘ 𝑇 ) ) |
| 160 |
159
|
ex |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ ∀ 𝑧 ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) → ( 𝑢 ∈ dom ( 𝑦 ∖ I ) → 𝑦 ∈ ( 𝐾 ‘ 𝑇 ) ) ) |
| 161 |
160
|
exlimdv |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ ∀ 𝑧 ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) → ( ∃ 𝑢 𝑢 ∈ dom ( 𝑦 ∖ I ) → 𝑦 ∈ ( 𝐾 ‘ 𝑇 ) ) ) |
| 162 |
55 161
|
syl5bi |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ ∀ 𝑧 ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) → ( dom ( 𝑦 ∖ I ) ≠ ∅ → 𝑦 ∈ ( 𝐾 ‘ 𝑇 ) ) ) |
| 163 |
54 162
|
pm2.61dne |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ ∀ 𝑧 ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) → 𝑦 ∈ ( 𝐾 ‘ 𝑇 ) ) |
| 164 |
163
|
exp31 |
⊢ ( dom ( 𝑦 ∖ I ) ∈ Fin → ( 𝑦 ∈ 𝐵 → ( ∀ 𝑧 ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) → 𝑦 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) |
| 165 |
164
|
com23 |
⊢ ( dom ( 𝑦 ∖ I ) ∈ Fin → ( ∀ 𝑧 ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) → ( 𝑦 ∈ 𝐵 → 𝑦 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) |
| 166 |
34 165
|
syl |
⊢ ( ( dom ( 𝑥 ∖ I ) ∈ Fin ∧ dom ( 𝑦 ∖ I ) ≼ dom ( 𝑥 ∖ I ) ) → ( ∀ 𝑧 ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) → ( 𝑦 ∈ 𝐵 → 𝑦 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) |
| 167 |
166
|
3impia |
⊢ ( ( dom ( 𝑥 ∖ I ) ∈ Fin ∧ dom ( 𝑦 ∖ I ) ≼ dom ( 𝑥 ∖ I ) ∧ ∀ 𝑧 ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) → ( 𝑦 ∈ 𝐵 → 𝑦 ∈ ( 𝐾 ‘ 𝑇 ) ) ) |
| 168 |
|
eleq1w |
⊢ ( 𝑦 = 𝑧 → ( 𝑦 ∈ 𝐵 ↔ 𝑧 ∈ 𝐵 ) ) |
| 169 |
|
eleq1w |
⊢ ( 𝑦 = 𝑧 → ( 𝑦 ∈ ( 𝐾 ‘ 𝑇 ) ↔ 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) |
| 170 |
168 169
|
imbi12d |
⊢ ( 𝑦 = 𝑧 → ( ( 𝑦 ∈ 𝐵 → 𝑦 ∈ ( 𝐾 ‘ 𝑇 ) ) ↔ ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) |
| 171 |
|
eleq1w |
⊢ ( 𝑦 = 𝑥 → ( 𝑦 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵 ) ) |
| 172 |
|
eleq1w |
⊢ ( 𝑦 = 𝑥 → ( 𝑦 ∈ ( 𝐾 ‘ 𝑇 ) ↔ 𝑥 ∈ ( 𝐾 ‘ 𝑇 ) ) ) |
| 173 |
171 172
|
imbi12d |
⊢ ( 𝑦 = 𝑥 → ( ( 𝑦 ∈ 𝐵 → 𝑦 ∈ ( 𝐾 ‘ 𝑇 ) ) ↔ ( 𝑥 ∈ 𝐵 → 𝑥 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) |
| 174 |
|
difeq1 |
⊢ ( 𝑦 = 𝑧 → ( 𝑦 ∖ I ) = ( 𝑧 ∖ I ) ) |
| 175 |
174
|
dmeqd |
⊢ ( 𝑦 = 𝑧 → dom ( 𝑦 ∖ I ) = dom ( 𝑧 ∖ I ) ) |
| 176 |
|
difeq1 |
⊢ ( 𝑦 = 𝑥 → ( 𝑦 ∖ I ) = ( 𝑥 ∖ I ) ) |
| 177 |
176
|
dmeqd |
⊢ ( 𝑦 = 𝑥 → dom ( 𝑦 ∖ I ) = dom ( 𝑥 ∖ I ) ) |
| 178 |
32 33 167 170 173 175 177
|
indcardi |
⊢ ( dom ( 𝑥 ∖ I ) ∈ Fin → ( 𝑥 ∈ 𝐵 → 𝑥 ∈ ( 𝐾 ‘ 𝑇 ) ) ) |
| 179 |
178
|
impcom |
⊢ ( ( 𝑥 ∈ 𝐵 ∧ dom ( 𝑥 ∖ I ) ∈ Fin ) → 𝑥 ∈ ( 𝐾 ‘ 𝑇 ) ) |
| 180 |
179
|
3adant1 |
⊢ ( ( 𝐷 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵 ∧ dom ( 𝑥 ∖ I ) ∈ Fin ) → 𝑥 ∈ ( 𝐾 ‘ 𝑇 ) ) |
| 181 |
180
|
rabssdv |
⊢ ( 𝐷 ∈ 𝑉 → { 𝑥 ∈ 𝐵 ∣ dom ( 𝑥 ∖ I ) ∈ Fin } ⊆ ( 𝐾 ‘ 𝑇 ) ) |
| 182 |
30 181
|
eqssd |
⊢ ( 𝐷 ∈ 𝑉 → ( 𝐾 ‘ 𝑇 ) = { 𝑥 ∈ 𝐵 ∣ dom ( 𝑥 ∖ I ) ∈ Fin } ) |