Step |
Hyp |
Ref |
Expression |
1 |
|
bnj1491.1 |
⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) } |
2 |
|
bnj1491.2 |
⊢ 𝑌 = 〈 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) 〉 |
3 |
|
bnj1491.3 |
⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) } |
4 |
|
bnj1491.4 |
⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ) |
5 |
|
bnj1491.5 |
⊢ 𝐷 = { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 } |
6 |
|
bnj1491.6 |
⊢ ( 𝜓 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ ) ) |
7 |
|
bnj1491.7 |
⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) ) |
8 |
|
bnj1491.8 |
⊢ ( 𝜏′ ↔ [ 𝑦 / 𝑥 ] 𝜏 ) |
9 |
|
bnj1491.9 |
⊢ 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ } |
10 |
|
bnj1491.10 |
⊢ 𝑃 = ∪ 𝐻 |
11 |
|
bnj1491.11 |
⊢ 𝑍 = 〈 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) 〉 |
12 |
|
bnj1491.12 |
⊢ 𝑄 = ( 𝑃 ∪ { 〈 𝑥 , ( 𝐺 ‘ 𝑍 ) 〉 } ) |
13 |
|
bnj1491.13 |
⊢ ( 𝜒 → ( 𝑄 ∈ 𝐶 ∧ dom 𝑄 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ) |
14 |
1 2 3 4 5 6 7 8 9 10 11 12
|
bnj1466 |
⊢ ( 𝑤 ∈ 𝑄 → ∀ 𝑓 𝑤 ∈ 𝑄 ) |
15 |
14
|
nfcii |
⊢ Ⅎ 𝑓 𝑄 |
16 |
3
|
bnj1317 |
⊢ ( 𝑤 ∈ 𝐶 → ∀ 𝑓 𝑤 ∈ 𝐶 ) |
17 |
16
|
nfcii |
⊢ Ⅎ 𝑓 𝐶 |
18 |
15 17
|
nfel |
⊢ Ⅎ 𝑓 𝑄 ∈ 𝐶 |
19 |
15
|
nfdm |
⊢ Ⅎ 𝑓 dom 𝑄 |
20 |
19
|
nfeq1 |
⊢ Ⅎ 𝑓 dom 𝑄 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) |
21 |
18 20
|
nfan |
⊢ Ⅎ 𝑓 ( 𝑄 ∈ 𝐶 ∧ dom 𝑄 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) |
22 |
|
eleq1 |
⊢ ( 𝑓 = 𝑄 → ( 𝑓 ∈ 𝐶 ↔ 𝑄 ∈ 𝐶 ) ) |
23 |
|
dmeq |
⊢ ( 𝑓 = 𝑄 → dom 𝑓 = dom 𝑄 ) |
24 |
23
|
eqeq1d |
⊢ ( 𝑓 = 𝑄 → ( dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ↔ dom 𝑄 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ) |
25 |
22 24
|
anbi12d |
⊢ ( 𝑓 = 𝑄 → ( ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ↔ ( 𝑄 ∈ 𝐶 ∧ dom 𝑄 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ) ) |
26 |
15 21 25
|
spcegf |
⊢ ( 𝑄 ∈ V → ( ( 𝑄 ∈ 𝐶 ∧ dom 𝑄 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) → ∃ 𝑓 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ) ) |
27 |
13 26
|
mpan9 |
⊢ ( ( 𝜒 ∧ 𝑄 ∈ V ) → ∃ 𝑓 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) ) |