Step |
Hyp |
Ref |
Expression |
1 |
|
bnj1312.1 |
⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) } |
2 |
|
bnj1312.2 |
⊢ 𝑌 = 〈 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) 〉 |
3 |
|
bnj1312.3 |
⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) } |
4 |
|
bnj1312.4 |
⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ) |
5 |
|
bnj1312.5 |
⊢ 𝐷 = { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 } |
6 |
|
bnj1312.6 |
⊢ ( 𝜓 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ ) ) |
7 |
|
bnj1312.7 |
⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) ) |
8 |
|
bnj1312.8 |
⊢ ( 𝜏′ ↔ [ 𝑦 / 𝑥 ] 𝜏 ) |
9 |
|
bnj1312.9 |
⊢ 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ } |
10 |
|
bnj1312.10 |
⊢ 𝑃 = ∪ 𝐻 |
11 |
|
bnj1312.11 |
⊢ 𝑍 = 〈 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) 〉 |
12 |
|
bnj1312.12 |
⊢ 𝑄 = ( 𝑃 ∪ { 〈 𝑥 , ( 𝐺 ‘ 𝑍 ) 〉 } ) |
13 |
|
bnj1312.13 |
⊢ 𝑊 = 〈 𝑧 , ( 𝑄 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) 〉 |
14 |
|
bnj1312.14 |
⊢ 𝐸 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) |
15 |
6
|
simplbi |
⊢ ( 𝜓 → 𝑅 FrSe 𝐴 ) |
16 |
5
|
ssrab3 |
⊢ 𝐷 ⊆ 𝐴 |
17 |
16
|
a1i |
⊢ ( 𝜓 → 𝐷 ⊆ 𝐴 ) |
18 |
6
|
simprbi |
⊢ ( 𝜓 → 𝐷 ≠ ∅ ) |
19 |
5
|
bnj1230 |
⊢ ( 𝑤 ∈ 𝐷 → ∀ 𝑥 𝑤 ∈ 𝐷 ) |
20 |
19
|
bnj1228 |
⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝐷 ⊆ 𝐴 ∧ 𝐷 ≠ ∅ ) → ∃ 𝑥 ∈ 𝐷 ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) |
21 |
15 17 18 20
|
syl3anc |
⊢ ( 𝜓 → ∃ 𝑥 ∈ 𝐷 ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) |
22 |
|
nfv |
⊢ Ⅎ 𝑥 𝑅 FrSe 𝐴 |
23 |
19
|
nfcii |
⊢ Ⅎ 𝑥 𝐷 |
24 |
|
nfcv |
⊢ Ⅎ 𝑥 ∅ |
25 |
23 24
|
nfne |
⊢ Ⅎ 𝑥 𝐷 ≠ ∅ |
26 |
22 25
|
nfan |
⊢ Ⅎ 𝑥 ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ ) |
27 |
6 26
|
nfxfr |
⊢ Ⅎ 𝑥 𝜓 |
28 |
27
|
nf5ri |
⊢ ( 𝜓 → ∀ 𝑥 𝜓 ) |
29 |
21 7 28
|
bnj1521 |
⊢ ( 𝜓 → ∃ 𝑥 𝜒 ) |
30 |
7
|
simp2bi |
⊢ ( 𝜒 → 𝑥 ∈ 𝐷 ) |
31 |
5
|
bnj1538 |
⊢ ( 𝑥 ∈ 𝐷 → ¬ ∃ 𝑓 𝜏 ) |
32 |
1 2 3 4 5 6 7 8 9 10 11 12
|
bnj1489 |
⊢ ( 𝜒 → 𝑄 ∈ V ) |
33 |
7 15
|
bnj835 |
⊢ ( 𝜒 → 𝑅 FrSe 𝐴 ) |
34 |
1 2 3 4 5 6 7 8 9 10
|
bnj1384 |
⊢ ( 𝑅 FrSe 𝐴 → Fun 𝑃 ) |
35 |
33 34
|
syl |
⊢ ( 𝜒 → Fun 𝑃 ) |
36 |
1 2 3 4 5 6 7 8 9 10
|
bnj1415 |
⊢ ( 𝜒 → dom 𝑃 = trCl ( 𝑥 , 𝐴 , 𝑅 ) ) |
37 |
35 36
|
bnj1422 |
⊢ ( 𝜒 → 𝑃 Fn trCl ( 𝑥 , 𝐴 , 𝑅 ) ) |
38 |
1 2 3 4 5 6 7 8 9 10 11 12 36
|
bnj1416 |
⊢ ( 𝜒 → dom 𝑄 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) |
39 |
1 2 3 4 5 6 7 8 9 10 11 12 35 38 36
|
bnj1421 |
⊢ ( 𝜒 → Fun 𝑄 ) |
40 |
39 38
|
bnj1422 |
⊢ ( 𝜒 → 𝑄 Fn ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) |
41 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 37 40
|
bnj1423 |
⊢ ( 𝜒 → ∀ 𝑧 ∈ 𝐸 ( 𝑄 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑊 ) ) |
42 |
14
|
fneq2i |
⊢ ( 𝑄 Fn 𝐸 ↔ 𝑄 Fn ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) |
43 |
40 42
|
sylibr |
⊢ ( 𝜒 → 𝑄 Fn 𝐸 ) |
44 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14
|
bnj1452 |
⊢ ( 𝜒 → 𝐸 ∈ 𝐵 ) |
45 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 32 41 43 44
|
bnj1463 |
⊢ ( 𝜒 → 𝑄 ∈ 𝐶 ) |
46 |
45 38
|
jca |
⊢ ( 𝜒 → ( 𝑄 ∈ 𝐶 ∧ dom 𝑄 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ) |
47 |
1 2 3 4 5 6 7 8 9 10 11 12 46
|
bnj1491 |
⊢ ( ( 𝜒 ∧ 𝑄 ∈ V ) → ∃ 𝑓 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ) |
48 |
32 47
|
mpdan |
⊢ ( 𝜒 → ∃ 𝑓 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ) |
49 |
48 4
|
bnj1198 |
⊢ ( 𝜒 → ∃ 𝑓 𝜏 ) |
50 |
31 49
|
nsyl3 |
⊢ ( 𝜒 → ¬ 𝑥 ∈ 𝐷 ) |
51 |
29 30 50
|
bnj1304 |
⊢ ¬ 𝜓 |
52 |
6 51
|
bnj1541 |
⊢ ( 𝑅 FrSe 𝐴 → 𝐷 = ∅ ) |
53 |
5 52
|
bnj1476 |
⊢ ( 𝑅 FrSe 𝐴 → ∀ 𝑥 ∈ 𝐴 ∃ 𝑓 𝜏 ) |
54 |
4
|
exbii |
⊢ ( ∃ 𝑓 𝜏 ↔ ∃ 𝑓 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ) |
55 |
|
df-rex |
⊢ ( ∃ 𝑓 ∈ 𝐶 dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ↔ ∃ 𝑓 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ) |
56 |
54 55
|
bitr4i |
⊢ ( ∃ 𝑓 𝜏 ↔ ∃ 𝑓 ∈ 𝐶 dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) |
57 |
56
|
ralbii |
⊢ ( ∀ 𝑥 ∈ 𝐴 ∃ 𝑓 𝜏 ↔ ∀ 𝑥 ∈ 𝐴 ∃ 𝑓 ∈ 𝐶 dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) |
58 |
53 57
|
sylib |
⊢ ( 𝑅 FrSe 𝐴 → ∀ 𝑥 ∈ 𝐴 ∃ 𝑓 ∈ 𝐶 dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) |