Step |
Hyp |
Ref |
Expression |
1 |
|
bnj1452.1 |
⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) } |
2 |
|
bnj1452.2 |
⊢ 𝑌 = 〈 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) 〉 |
3 |
|
bnj1452.3 |
⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) } |
4 |
|
bnj1452.4 |
⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ) |
5 |
|
bnj1452.5 |
⊢ 𝐷 = { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 } |
6 |
|
bnj1452.6 |
⊢ ( 𝜓 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ ) ) |
7 |
|
bnj1452.7 |
⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) ) |
8 |
|
bnj1452.8 |
⊢ ( 𝜏′ ↔ [ 𝑦 / 𝑥 ] 𝜏 ) |
9 |
|
bnj1452.9 |
⊢ 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ } |
10 |
|
bnj1452.10 |
⊢ 𝑃 = ∪ 𝐻 |
11 |
|
bnj1452.11 |
⊢ 𝑍 = 〈 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) 〉 |
12 |
|
bnj1452.12 |
⊢ 𝑄 = ( 𝑃 ∪ { 〈 𝑥 , ( 𝐺 ‘ 𝑍 ) 〉 } ) |
13 |
|
bnj1452.13 |
⊢ 𝑊 = 〈 𝑧 , ( 𝑄 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) 〉 |
14 |
|
bnj1452.14 |
⊢ 𝐸 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) |
15 |
5 7
|
bnj1212 |
⊢ ( 𝜒 → 𝑥 ∈ 𝐴 ) |
16 |
15
|
snssd |
⊢ ( 𝜒 → { 𝑥 } ⊆ 𝐴 ) |
17 |
|
bnj1147 |
⊢ trCl ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝐴 |
18 |
17
|
a1i |
⊢ ( 𝜒 → trCl ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝐴 ) |
19 |
16 18
|
unssd |
⊢ ( 𝜒 → ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ⊆ 𝐴 ) |
20 |
14 19
|
eqsstrid |
⊢ ( 𝜒 → 𝐸 ⊆ 𝐴 ) |
21 |
|
elsni |
⊢ ( 𝑧 ∈ { 𝑥 } → 𝑧 = 𝑥 ) |
22 |
21
|
adantl |
⊢ ( ( ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) ∧ 𝑧 ∈ { 𝑥 } ) → 𝑧 = 𝑥 ) |
23 |
|
bnj602 |
⊢ ( 𝑧 = 𝑥 → pred ( 𝑧 , 𝐴 , 𝑅 ) = pred ( 𝑥 , 𝐴 , 𝑅 ) ) |
24 |
22 23
|
syl |
⊢ ( ( ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) ∧ 𝑧 ∈ { 𝑥 } ) → pred ( 𝑧 , 𝐴 , 𝑅 ) = pred ( 𝑥 , 𝐴 , 𝑅 ) ) |
25 |
6
|
simplbi |
⊢ ( 𝜓 → 𝑅 FrSe 𝐴 ) |
26 |
7 25
|
bnj835 |
⊢ ( 𝜒 → 𝑅 FrSe 𝐴 ) |
27 |
|
bnj906 |
⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) |
28 |
26 15 27
|
syl2anc |
⊢ ( 𝜒 → pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) |
29 |
28
|
ad2antrr |
⊢ ( ( ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) ∧ 𝑧 ∈ { 𝑥 } ) → pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) |
30 |
24 29
|
eqsstrd |
⊢ ( ( ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) ∧ 𝑧 ∈ { 𝑥 } ) → pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) |
31 |
|
ssun4 |
⊢ ( pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑥 , 𝐴 , 𝑅 ) → pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) |
32 |
31 14
|
sseqtrrdi |
⊢ ( pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑥 , 𝐴 , 𝑅 ) → pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝐸 ) |
33 |
30 32
|
syl |
⊢ ( ( ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) ∧ 𝑧 ∈ { 𝑥 } ) → pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝐸 ) |
34 |
26
|
ad2antrr |
⊢ ( ( ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) ∧ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) → 𝑅 FrSe 𝐴 ) |
35 |
|
simpr |
⊢ ( ( ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) ∧ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) → 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) |
36 |
17 35
|
bnj1213 |
⊢ ( ( ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) ∧ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) → 𝑧 ∈ 𝐴 ) |
37 |
|
bnj906 |
⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴 ) → pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑧 , 𝐴 , 𝑅 ) ) |
38 |
34 36 37
|
syl2anc |
⊢ ( ( ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) ∧ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) → pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑧 , 𝐴 , 𝑅 ) ) |
39 |
15
|
ad2antrr |
⊢ ( ( ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) ∧ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) → 𝑥 ∈ 𝐴 ) |
40 |
|
bnj1125 |
⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) → trCl ( 𝑧 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) |
41 |
34 39 35 40
|
syl3anc |
⊢ ( ( ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) ∧ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) → trCl ( 𝑧 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) |
42 |
38 41
|
sstrd |
⊢ ( ( ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) ∧ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) → pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) |
43 |
42 32
|
syl |
⊢ ( ( ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) ∧ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) → pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝐸 ) |
44 |
14
|
bnj1424 |
⊢ ( 𝑧 ∈ 𝐸 → ( 𝑧 ∈ { 𝑥 } ∨ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) |
45 |
44
|
adantl |
⊢ ( ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) → ( 𝑧 ∈ { 𝑥 } ∨ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) |
46 |
33 43 45
|
mpjaodan |
⊢ ( ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) → pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝐸 ) |
47 |
46
|
ralrimiva |
⊢ ( 𝜒 → ∀ 𝑧 ∈ 𝐸 pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝐸 ) |
48 |
|
snex |
⊢ { 𝑥 } ∈ V |
49 |
48
|
a1i |
⊢ ( 𝜒 → { 𝑥 } ∈ V ) |
50 |
|
bnj893 |
⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → trCl ( 𝑥 , 𝐴 , 𝑅 ) ∈ V ) |
51 |
26 15 50
|
syl2anc |
⊢ ( 𝜒 → trCl ( 𝑥 , 𝐴 , 𝑅 ) ∈ V ) |
52 |
49 51
|
bnj1149 |
⊢ ( 𝜒 → ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ∈ V ) |
53 |
14 52
|
eqeltrid |
⊢ ( 𝜒 → 𝐸 ∈ V ) |
54 |
1
|
bnj1454 |
⊢ ( 𝐸 ∈ V → ( 𝐸 ∈ 𝐵 ↔ [ 𝐸 / 𝑑 ] ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) ) ) |
55 |
53 54
|
syl |
⊢ ( 𝜒 → ( 𝐸 ∈ 𝐵 ↔ [ 𝐸 / 𝑑 ] ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) ) ) |
56 |
|
bnj602 |
⊢ ( 𝑥 = 𝑧 → pred ( 𝑥 , 𝐴 , 𝑅 ) = pred ( 𝑧 , 𝐴 , 𝑅 ) ) |
57 |
56
|
sseq1d |
⊢ ( 𝑥 = 𝑧 → ( pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ↔ pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) ) |
58 |
57
|
cbvralvw |
⊢ ( ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ↔ ∀ 𝑧 ∈ 𝑑 pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) |
59 |
58
|
anbi2i |
⊢ ( ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) ↔ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝑑 pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) ) |
60 |
59
|
sbcbii |
⊢ ( [ 𝐸 / 𝑑 ] ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) ↔ [ 𝐸 / 𝑑 ] ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝑑 pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) ) |
61 |
55 60
|
bitrdi |
⊢ ( 𝜒 → ( 𝐸 ∈ 𝐵 ↔ [ 𝐸 / 𝑑 ] ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝑑 pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) ) ) |
62 |
|
sseq1 |
⊢ ( 𝑑 = 𝐸 → ( 𝑑 ⊆ 𝐴 ↔ 𝐸 ⊆ 𝐴 ) ) |
63 |
|
sseq2 |
⊢ ( 𝑑 = 𝐸 → ( pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝑑 ↔ pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝐸 ) ) |
64 |
63
|
raleqbi1dv |
⊢ ( 𝑑 = 𝐸 → ( ∀ 𝑧 ∈ 𝑑 pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝑑 ↔ ∀ 𝑧 ∈ 𝐸 pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝐸 ) ) |
65 |
62 64
|
anbi12d |
⊢ ( 𝑑 = 𝐸 → ( ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝑑 pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) ↔ ( 𝐸 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐸 pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝐸 ) ) ) |
66 |
65
|
sbcieg |
⊢ ( 𝐸 ∈ V → ( [ 𝐸 / 𝑑 ] ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝑑 pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) ↔ ( 𝐸 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐸 pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝐸 ) ) ) |
67 |
53 66
|
syl |
⊢ ( 𝜒 → ( [ 𝐸 / 𝑑 ] ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝑑 pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) ↔ ( 𝐸 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐸 pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝐸 ) ) ) |
68 |
61 67
|
bitrd |
⊢ ( 𝜒 → ( 𝐸 ∈ 𝐵 ↔ ( 𝐸 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐸 pred ( 𝑧 , 𝐴 , 𝑅 ) ⊆ 𝐸 ) ) ) |
69 |
20 47 68
|
mpbir2and |
⊢ ( 𝜒 → 𝐸 ∈ 𝐵 ) |