Step |
Hyp |
Ref |
Expression |
1 |
|
bnj1489.1 |
⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) } |
2 |
|
bnj1489.2 |
⊢ 𝑌 = 〈 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) 〉 |
3 |
|
bnj1489.3 |
⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) } |
4 |
|
bnj1489.4 |
⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ) |
5 |
|
bnj1489.5 |
⊢ 𝐷 = { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 } |
6 |
|
bnj1489.6 |
⊢ ( 𝜓 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ ) ) |
7 |
|
bnj1489.7 |
⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) ) |
8 |
|
bnj1489.8 |
⊢ ( 𝜏′ ↔ [ 𝑦 / 𝑥 ] 𝜏 ) |
9 |
|
bnj1489.9 |
⊢ 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ } |
10 |
|
bnj1489.10 |
⊢ 𝑃 = ∪ 𝐻 |
11 |
|
bnj1489.11 |
⊢ 𝑍 = 〈 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) 〉 |
12 |
|
bnj1489.12 |
⊢ 𝑄 = ( 𝑃 ∪ { 〈 𝑥 , ( 𝐺 ‘ 𝑍 ) 〉 } ) |
13 |
|
bnj1364 |
⊢ ( 𝑅 FrSe 𝐴 → 𝑅 Se 𝐴 ) |
14 |
|
df-bnj13 |
⊢ ( 𝑅 Se 𝐴 ↔ ∀ 𝑥 ∈ 𝐴 pred ( 𝑥 , 𝐴 , 𝑅 ) ∈ V ) |
15 |
13 14
|
sylib |
⊢ ( 𝑅 FrSe 𝐴 → ∀ 𝑥 ∈ 𝐴 pred ( 𝑥 , 𝐴 , 𝑅 ) ∈ V ) |
16 |
6 15
|
bnj832 |
⊢ ( 𝜓 → ∀ 𝑥 ∈ 𝐴 pred ( 𝑥 , 𝐴 , 𝑅 ) ∈ V ) |
17 |
7 16
|
bnj835 |
⊢ ( 𝜒 → ∀ 𝑥 ∈ 𝐴 pred ( 𝑥 , 𝐴 , 𝑅 ) ∈ V ) |
18 |
5 7
|
bnj1212 |
⊢ ( 𝜒 → 𝑥 ∈ 𝐴 ) |
19 |
17 18
|
bnj1294 |
⊢ ( 𝜒 → pred ( 𝑥 , 𝐴 , 𝑅 ) ∈ V ) |
20 |
|
nfv |
⊢ Ⅎ 𝑦 𝜓 |
21 |
|
nfv |
⊢ Ⅎ 𝑦 𝑥 ∈ 𝐷 |
22 |
|
nfra1 |
⊢ Ⅎ 𝑦 ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 |
23 |
20 21 22
|
nf3an |
⊢ Ⅎ 𝑦 ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) |
24 |
7 23
|
nfxfr |
⊢ Ⅎ 𝑦 𝜒 |
25 |
6
|
simplbi |
⊢ ( 𝜓 → 𝑅 FrSe 𝐴 ) |
26 |
7 25
|
bnj835 |
⊢ ( 𝜒 → 𝑅 FrSe 𝐴 ) |
27 |
26
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ) → 𝑅 FrSe 𝐴 ) |
28 |
1 2 3 4 5 6 7 8
|
bnj1388 |
⊢ ( 𝜒 → ∀ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ∃ 𝑓 𝜏′ ) |
29 |
28
|
r19.21bi |
⊢ ( ( 𝜒 ∧ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ) → ∃ 𝑓 𝜏′ ) |
30 |
|
nfv |
⊢ Ⅎ 𝑥 𝑅 FrSe 𝐴 |
31 |
|
nfsbc1v |
⊢ Ⅎ 𝑥 [ 𝑦 / 𝑥 ] 𝜏 |
32 |
8 31
|
nfxfr |
⊢ Ⅎ 𝑥 𝜏′ |
33 |
32
|
nfex |
⊢ Ⅎ 𝑥 ∃ 𝑓 𝜏′ |
34 |
30 33
|
nfan |
⊢ Ⅎ 𝑥 ( 𝑅 FrSe 𝐴 ∧ ∃ 𝑓 𝜏′ ) |
35 |
32
|
nfeuw |
⊢ Ⅎ 𝑥 ∃! 𝑓 𝜏′ |
36 |
34 35
|
nfim |
⊢ Ⅎ 𝑥 ( ( 𝑅 FrSe 𝐴 ∧ ∃ 𝑓 𝜏′ ) → ∃! 𝑓 𝜏′ ) |
37 |
|
sneq |
⊢ ( 𝑥 = 𝑦 → { 𝑥 } = { 𝑦 } ) |
38 |
|
bnj1318 |
⊢ ( 𝑥 = 𝑦 → trCl ( 𝑥 , 𝐴 , 𝑅 ) = trCl ( 𝑦 , 𝐴 , 𝑅 ) ) |
39 |
37 38
|
uneq12d |
⊢ ( 𝑥 = 𝑦 → ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) ) |
40 |
39
|
eqeq2d |
⊢ ( 𝑥 = 𝑦 → ( dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ↔ dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) ) ) |
41 |
40
|
anbi2d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) ) ) ) |
42 |
1 2 3 4 8
|
bnj1373 |
⊢ ( 𝜏′ ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) ) ) |
43 |
41 42
|
bitr4di |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ↔ 𝜏′ ) ) |
44 |
43
|
exbidv |
⊢ ( 𝑥 = 𝑦 → ( ∃ 𝑓 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ↔ ∃ 𝑓 𝜏′ ) ) |
45 |
44
|
anbi2d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑅 FrSe 𝐴 ∧ ∃ 𝑓 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ) ↔ ( 𝑅 FrSe 𝐴 ∧ ∃ 𝑓 𝜏′ ) ) ) |
46 |
43
|
eubidv |
⊢ ( 𝑥 = 𝑦 → ( ∃! 𝑓 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ↔ ∃! 𝑓 𝜏′ ) ) |
47 |
45 46
|
imbi12d |
⊢ ( 𝑥 = 𝑦 → ( ( ( 𝑅 FrSe 𝐴 ∧ ∃ 𝑓 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ) → ∃! 𝑓 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ ∃ 𝑓 𝜏′ ) → ∃! 𝑓 𝜏′ ) ) ) |
48 |
|
biid |
⊢ ( ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ) |
49 |
1 2 3 48
|
bnj1321 |
⊢ ( ( 𝑅 FrSe 𝐴 ∧ ∃ 𝑓 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ) → ∃! 𝑓 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ) |
50 |
36 47 49
|
chvarfv |
⊢ ( ( 𝑅 FrSe 𝐴 ∧ ∃ 𝑓 𝜏′ ) → ∃! 𝑓 𝜏′ ) |
51 |
27 29 50
|
syl2anc |
⊢ ( ( 𝜒 ∧ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ) → ∃! 𝑓 𝜏′ ) |
52 |
51
|
ex |
⊢ ( 𝜒 → ( 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) → ∃! 𝑓 𝜏′ ) ) |
53 |
24 52
|
ralrimi |
⊢ ( 𝜒 → ∀ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ∃! 𝑓 𝜏′ ) |
54 |
9
|
a1i |
⊢ ( 𝜒 → 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ } ) |
55 |
|
biid |
⊢ ( ( pred ( 𝑥 , 𝐴 , 𝑅 ) ∈ V ∧ ∀ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ∃! 𝑓 𝜏′ ∧ 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ } ) ↔ ( pred ( 𝑥 , 𝐴 , 𝑅 ) ∈ V ∧ ∀ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ∃! 𝑓 𝜏′ ∧ 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ } ) ) |
56 |
55
|
bnj1366 |
⊢ ( ( pred ( 𝑥 , 𝐴 , 𝑅 ) ∈ V ∧ ∀ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ∃! 𝑓 𝜏′ ∧ 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ } ) → 𝐻 ∈ V ) |
57 |
19 53 54 56
|
syl3anc |
⊢ ( 𝜒 → 𝐻 ∈ V ) |
58 |
57
|
uniexd |
⊢ ( 𝜒 → ∪ 𝐻 ∈ V ) |
59 |
10 58
|
eqeltrid |
⊢ ( 𝜒 → 𝑃 ∈ V ) |
60 |
|
snex |
⊢ { 〈 𝑥 , ( 𝐺 ‘ 𝑍 ) 〉 } ∈ V |
61 |
60
|
a1i |
⊢ ( 𝜒 → { 〈 𝑥 , ( 𝐺 ‘ 𝑍 ) 〉 } ∈ V ) |
62 |
59 61
|
bnj1149 |
⊢ ( 𝜒 → ( 𝑃 ∪ { 〈 𝑥 , ( 𝐺 ‘ 𝑍 ) 〉 } ) ∈ V ) |
63 |
12 62
|
eqeltrid |
⊢ ( 𝜒 → 𝑄 ∈ V ) |