Step |
Hyp |
Ref |
Expression |
1 |
|
bnj1525.1 |
⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) } |
2 |
|
bnj1525.2 |
⊢ 𝑌 = 〈 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) 〉 |
3 |
|
bnj1525.3 |
⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) } |
4 |
|
bnj1525.4 |
⊢ 𝐹 = ∪ 𝐶 |
5 |
|
bnj1525.5 |
⊢ ( 𝜑 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐻 Fn 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐻 ‘ 𝑥 ) = ( 𝐺 ‘ 〈 𝑥 , ( 𝐻 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) 〉 ) ) ) |
6 |
|
bnj1525.6 |
⊢ ( 𝜓 ↔ ( 𝜑 ∧ 𝐹 ≠ 𝐻 ) ) |
7 |
|
nfv |
⊢ Ⅎ 𝑥 𝑅 FrSe 𝐴 |
8 |
|
nfv |
⊢ Ⅎ 𝑥 𝐻 Fn 𝐴 |
9 |
|
nfra1 |
⊢ Ⅎ 𝑥 ∀ 𝑥 ∈ 𝐴 ( 𝐻 ‘ 𝑥 ) = ( 𝐺 ‘ 〈 𝑥 , ( 𝐻 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) 〉 ) |
10 |
7 8 9
|
nf3an |
⊢ Ⅎ 𝑥 ( 𝑅 FrSe 𝐴 ∧ 𝐻 Fn 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐻 ‘ 𝑥 ) = ( 𝐺 ‘ 〈 𝑥 , ( 𝐻 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) 〉 ) ) |
11 |
5 10
|
nfxfr |
⊢ Ⅎ 𝑥 𝜑 |
12 |
1
|
bnj1309 |
⊢ ( 𝑤 ∈ 𝐵 → ∀ 𝑥 𝑤 ∈ 𝐵 ) |
13 |
3 12
|
bnj1307 |
⊢ ( 𝑤 ∈ 𝐶 → ∀ 𝑥 𝑤 ∈ 𝐶 ) |
14 |
13
|
nfcii |
⊢ Ⅎ 𝑥 𝐶 |
15 |
14
|
nfuni |
⊢ Ⅎ 𝑥 ∪ 𝐶 |
16 |
4 15
|
nfcxfr |
⊢ Ⅎ 𝑥 𝐹 |
17 |
|
nfcv |
⊢ Ⅎ 𝑥 𝐻 |
18 |
16 17
|
nfne |
⊢ Ⅎ 𝑥 𝐹 ≠ 𝐻 |
19 |
11 18
|
nfan |
⊢ Ⅎ 𝑥 ( 𝜑 ∧ 𝐹 ≠ 𝐻 ) |
20 |
6 19
|
nfxfr |
⊢ Ⅎ 𝑥 𝜓 |
21 |
20
|
nf5ri |
⊢ ( 𝜓 → ∀ 𝑥 𝜓 ) |