Step |
Hyp |
Ref |
Expression |
1 |
|
bnj1445.1 |
⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) } |
2 |
|
bnj1445.2 |
⊢ 𝑌 = 〈 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) 〉 |
3 |
|
bnj1445.3 |
⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) } |
4 |
|
bnj1445.4 |
⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ) |
5 |
|
bnj1445.5 |
⊢ 𝐷 = { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 } |
6 |
|
bnj1445.6 |
⊢ ( 𝜓 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ ) ) |
7 |
|
bnj1445.7 |
⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) ) |
8 |
|
bnj1445.8 |
⊢ ( 𝜏′ ↔ [ 𝑦 / 𝑥 ] 𝜏 ) |
9 |
|
bnj1445.9 |
⊢ 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ } |
10 |
|
bnj1445.10 |
⊢ 𝑃 = ∪ 𝐻 |
11 |
|
bnj1445.11 |
⊢ 𝑍 = 〈 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) 〉 |
12 |
|
bnj1445.12 |
⊢ 𝑄 = ( 𝑃 ∪ { 〈 𝑥 , ( 𝐺 ‘ 𝑍 ) 〉 } ) |
13 |
|
bnj1445.13 |
⊢ 𝑊 = 〈 𝑧 , ( 𝑄 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) 〉 |
14 |
|
bnj1445.14 |
⊢ 𝐸 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) |
15 |
|
bnj1445.15 |
⊢ ( 𝜒 → 𝑃 Fn trCl ( 𝑥 , 𝐴 , 𝑅 ) ) |
16 |
|
bnj1445.16 |
⊢ ( 𝜒 → 𝑄 Fn ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) |
17 |
|
bnj1445.17 |
⊢ ( 𝜃 ↔ ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) ) |
18 |
|
bnj1445.18 |
⊢ ( 𝜂 ↔ ( 𝜃 ∧ 𝑧 ∈ { 𝑥 } ) ) |
19 |
|
bnj1445.19 |
⊢ ( 𝜁 ↔ ( 𝜃 ∧ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) |
20 |
|
bnj1445.20 |
⊢ ( 𝜌 ↔ ( 𝜁 ∧ 𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓 ) ) |
21 |
|
bnj1445.21 |
⊢ ( 𝜎 ↔ ( 𝜌 ∧ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ∧ 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) ) ) |
22 |
|
bnj1445.22 |
⊢ ( 𝜑 ↔ ( 𝜎 ∧ 𝑑 ∈ 𝐵 ∧ 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) ) |
23 |
|
bnj1445.23 |
⊢ 𝑋 = 〈 𝑧 , ( 𝑓 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) 〉 |
24 |
|
nfv |
⊢ Ⅎ 𝑑 𝑅 FrSe 𝐴 |
25 |
|
nfre1 |
⊢ Ⅎ 𝑑 ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) |
26 |
25
|
nfab |
⊢ Ⅎ 𝑑 { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) } |
27 |
3 26
|
nfcxfr |
⊢ Ⅎ 𝑑 𝐶 |
28 |
27
|
nfcri |
⊢ Ⅎ 𝑑 𝑓 ∈ 𝐶 |
29 |
|
nfv |
⊢ Ⅎ 𝑑 dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) |
30 |
28 29
|
nfan |
⊢ Ⅎ 𝑑 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) |
31 |
4 30
|
nfxfr |
⊢ Ⅎ 𝑑 𝜏 |
32 |
31
|
nfex |
⊢ Ⅎ 𝑑 ∃ 𝑓 𝜏 |
33 |
32
|
nfn |
⊢ Ⅎ 𝑑 ¬ ∃ 𝑓 𝜏 |
34 |
|
nfcv |
⊢ Ⅎ 𝑑 𝐴 |
35 |
33 34
|
nfrabw |
⊢ Ⅎ 𝑑 { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 } |
36 |
5 35
|
nfcxfr |
⊢ Ⅎ 𝑑 𝐷 |
37 |
|
nfcv |
⊢ Ⅎ 𝑑 ∅ |
38 |
36 37
|
nfne |
⊢ Ⅎ 𝑑 𝐷 ≠ ∅ |
39 |
24 38
|
nfan |
⊢ Ⅎ 𝑑 ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ ) |
40 |
6 39
|
nfxfr |
⊢ Ⅎ 𝑑 𝜓 |
41 |
36
|
nfcri |
⊢ Ⅎ 𝑑 𝑥 ∈ 𝐷 |
42 |
|
nfv |
⊢ Ⅎ 𝑑 ¬ 𝑦 𝑅 𝑥 |
43 |
36 42
|
nfralw |
⊢ Ⅎ 𝑑 ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 |
44 |
40 41 43
|
nf3an |
⊢ Ⅎ 𝑑 ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) |
45 |
7 44
|
nfxfr |
⊢ Ⅎ 𝑑 𝜒 |
46 |
45
|
nf5ri |
⊢ ( 𝜒 → ∀ 𝑑 𝜒 ) |
47 |
46
|
bnj1351 |
⊢ ( ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) → ∀ 𝑑 ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) ) |
48 |
47
|
nf5i |
⊢ Ⅎ 𝑑 ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) |
49 |
17 48
|
nfxfr |
⊢ Ⅎ 𝑑 𝜃 |
50 |
|
nfv |
⊢ Ⅎ 𝑑 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) |
51 |
49 50
|
nfan |
⊢ Ⅎ 𝑑 ( 𝜃 ∧ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) |
52 |
19 51
|
nfxfr |
⊢ Ⅎ 𝑑 𝜁 |
53 |
|
nfcv |
⊢ Ⅎ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) |
54 |
|
nfcv |
⊢ Ⅎ 𝑑 𝑦 |
55 |
54 31
|
nfsbcw |
⊢ Ⅎ 𝑑 [ 𝑦 / 𝑥 ] 𝜏 |
56 |
8 55
|
nfxfr |
⊢ Ⅎ 𝑑 𝜏′ |
57 |
53 56
|
nfrex |
⊢ Ⅎ 𝑑 ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ |
58 |
57
|
nfab |
⊢ Ⅎ 𝑑 { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ } |
59 |
9 58
|
nfcxfr |
⊢ Ⅎ 𝑑 𝐻 |
60 |
59
|
nfcri |
⊢ Ⅎ 𝑑 𝑓 ∈ 𝐻 |
61 |
|
nfv |
⊢ Ⅎ 𝑑 𝑧 ∈ dom 𝑓 |
62 |
52 60 61
|
nf3an |
⊢ Ⅎ 𝑑 ( 𝜁 ∧ 𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓 ) |
63 |
20 62
|
nfxfr |
⊢ Ⅎ 𝑑 𝜌 |
64 |
63
|
nf5ri |
⊢ ( 𝜌 → ∀ 𝑑 𝜌 ) |
65 |
|
ax-5 |
⊢ ( 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) → ∀ 𝑑 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ) |
66 |
28
|
nf5ri |
⊢ ( 𝑓 ∈ 𝐶 → ∀ 𝑑 𝑓 ∈ 𝐶 ) |
67 |
|
ax-5 |
⊢ ( dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) → ∀ 𝑑 dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) ) |
68 |
64 65 66 67
|
bnj982 |
⊢ ( ( 𝜌 ∧ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ∧ 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) ) → ∀ 𝑑 ( 𝜌 ∧ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ∧ 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) ) ) |
69 |
21 68
|
hbxfrbi |
⊢ ( 𝜎 → ∀ 𝑑 𝜎 ) |