Step |
Hyp |
Ref |
Expression |
1 |
|
bnj1421.1 |
⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) } |
2 |
|
bnj1421.2 |
⊢ 𝑌 = 〈 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) 〉 |
3 |
|
bnj1421.3 |
⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) } |
4 |
|
bnj1421.4 |
⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ) |
5 |
|
bnj1421.5 |
⊢ 𝐷 = { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 } |
6 |
|
bnj1421.6 |
⊢ ( 𝜓 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ ) ) |
7 |
|
bnj1421.7 |
⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) ) |
8 |
|
bnj1421.8 |
⊢ ( 𝜏′ ↔ [ 𝑦 / 𝑥 ] 𝜏 ) |
9 |
|
bnj1421.9 |
⊢ 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ } |
10 |
|
bnj1421.10 |
⊢ 𝑃 = ∪ 𝐻 |
11 |
|
bnj1421.11 |
⊢ 𝑍 = 〈 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) 〉 |
12 |
|
bnj1421.12 |
⊢ 𝑄 = ( 𝑃 ∪ { 〈 𝑥 , ( 𝐺 ‘ 𝑍 ) 〉 } ) |
13 |
|
bnj1421.13 |
⊢ ( 𝜒 → Fun 𝑃 ) |
14 |
|
bnj1421.14 |
⊢ ( 𝜒 → dom 𝑄 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) |
15 |
|
bnj1421.15 |
⊢ ( 𝜒 → dom 𝑃 = trCl ( 𝑥 , 𝐴 , 𝑅 ) ) |
16 |
|
vex |
⊢ 𝑥 ∈ V |
17 |
|
fvex |
⊢ ( 𝐺 ‘ 𝑍 ) ∈ V |
18 |
16 17
|
funsn |
⊢ Fun { 〈 𝑥 , ( 𝐺 ‘ 𝑍 ) 〉 } |
19 |
13 18
|
jctir |
⊢ ( 𝜒 → ( Fun 𝑃 ∧ Fun { 〈 𝑥 , ( 𝐺 ‘ 𝑍 ) 〉 } ) ) |
20 |
17
|
dmsnop |
⊢ dom { 〈 𝑥 , ( 𝐺 ‘ 𝑍 ) 〉 } = { 𝑥 } |
21 |
20
|
a1i |
⊢ ( 𝜒 → dom { 〈 𝑥 , ( 𝐺 ‘ 𝑍 ) 〉 } = { 𝑥 } ) |
22 |
15 21
|
ineq12d |
⊢ ( 𝜒 → ( dom 𝑃 ∩ dom { 〈 𝑥 , ( 𝐺 ‘ 𝑍 ) 〉 } ) = ( trCl ( 𝑥 , 𝐴 , 𝑅 ) ∩ { 𝑥 } ) ) |
23 |
6
|
simplbi |
⊢ ( 𝜓 → 𝑅 FrSe 𝐴 ) |
24 |
7 23
|
bnj835 |
⊢ ( 𝜒 → 𝑅 FrSe 𝐴 ) |
25 |
|
biid |
⊢ ( 𝑅 FrSe 𝐴 ↔ 𝑅 FrSe 𝐴 ) |
26 |
|
biid |
⊢ ( ¬ 𝑥 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ↔ ¬ 𝑥 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) |
27 |
|
biid |
⊢ ( ∀ 𝑧 ∈ 𝐴 ( 𝑧 𝑅 𝑥 → [ 𝑧 / 𝑥 ] ¬ 𝑥 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ↔ ∀ 𝑧 ∈ 𝐴 ( 𝑧 𝑅 𝑥 → [ 𝑧 / 𝑥 ] ¬ 𝑥 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) |
28 |
|
biid |
⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ( 𝑧 𝑅 𝑥 → [ 𝑧 / 𝑥 ] ¬ 𝑥 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ↔ ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ( 𝑧 𝑅 𝑥 → [ 𝑧 / 𝑥 ] ¬ 𝑥 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ) |
29 |
|
eqid |
⊢ ( pred ( 𝑥 , 𝐴 , 𝑅 ) ∪ ∪ 𝑧 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) trCl ( 𝑧 , 𝐴 , 𝑅 ) ) = ( pred ( 𝑥 , 𝐴 , 𝑅 ) ∪ ∪ 𝑧 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) trCl ( 𝑧 , 𝐴 , 𝑅 ) ) |
30 |
25 26 27 28 29
|
bnj1417 |
⊢ ( 𝑅 FrSe 𝐴 → ∀ 𝑥 ∈ 𝐴 ¬ 𝑥 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) |
31 |
|
disjsn |
⊢ ( ( trCl ( 𝑥 , 𝐴 , 𝑅 ) ∩ { 𝑥 } ) = ∅ ↔ ¬ 𝑥 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) |
32 |
31
|
ralbii |
⊢ ( ∀ 𝑥 ∈ 𝐴 ( trCl ( 𝑥 , 𝐴 , 𝑅 ) ∩ { 𝑥 } ) = ∅ ↔ ∀ 𝑥 ∈ 𝐴 ¬ 𝑥 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) |
33 |
30 32
|
sylibr |
⊢ ( 𝑅 FrSe 𝐴 → ∀ 𝑥 ∈ 𝐴 ( trCl ( 𝑥 , 𝐴 , 𝑅 ) ∩ { 𝑥 } ) = ∅ ) |
34 |
24 33
|
syl |
⊢ ( 𝜒 → ∀ 𝑥 ∈ 𝐴 ( trCl ( 𝑥 , 𝐴 , 𝑅 ) ∩ { 𝑥 } ) = ∅ ) |
35 |
5 7
|
bnj1212 |
⊢ ( 𝜒 → 𝑥 ∈ 𝐴 ) |
36 |
34 35
|
bnj1294 |
⊢ ( 𝜒 → ( trCl ( 𝑥 , 𝐴 , 𝑅 ) ∩ { 𝑥 } ) = ∅ ) |
37 |
22 36
|
eqtrd |
⊢ ( 𝜒 → ( dom 𝑃 ∩ dom { 〈 𝑥 , ( 𝐺 ‘ 𝑍 ) 〉 } ) = ∅ ) |
38 |
|
funun |
⊢ ( ( ( Fun 𝑃 ∧ Fun { 〈 𝑥 , ( 𝐺 ‘ 𝑍 ) 〉 } ) ∧ ( dom 𝑃 ∩ dom { 〈 𝑥 , ( 𝐺 ‘ 𝑍 ) 〉 } ) = ∅ ) → Fun ( 𝑃 ∪ { 〈 𝑥 , ( 𝐺 ‘ 𝑍 ) 〉 } ) ) |
39 |
19 37 38
|
syl2anc |
⊢ ( 𝜒 → Fun ( 𝑃 ∪ { 〈 𝑥 , ( 𝐺 ‘ 𝑍 ) 〉 } ) ) |
40 |
12
|
funeqi |
⊢ ( Fun 𝑄 ↔ Fun ( 𝑃 ∪ { 〈 𝑥 , ( 𝐺 ‘ 𝑍 ) 〉 } ) ) |
41 |
39 40
|
sylibr |
⊢ ( 𝜒 → Fun 𝑄 ) |