Step |
Hyp |
Ref |
Expression |
1 |
|
bnj130.1 |
⊢ ( 𝜃 ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ∃! 𝑓 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) ) |
2 |
|
bnj130.2 |
⊢ ( 𝜑′ ↔ [ 1o / 𝑛 ] 𝜑 ) |
3 |
|
bnj130.3 |
⊢ ( 𝜓′ ↔ [ 1o / 𝑛 ] 𝜓 ) |
4 |
|
bnj130.4 |
⊢ ( 𝜃′ ↔ [ 1o / 𝑛 ] 𝜃 ) |
5 |
1
|
sbcbii |
⊢ ( [ 1o / 𝑛 ] 𝜃 ↔ [ 1o / 𝑛 ] ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ∃! 𝑓 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) ) |
6 |
|
bnj105 |
⊢ 1o ∈ V |
7 |
6
|
bnj90 |
⊢ ( [ 1o / 𝑛 ] 𝑓 Fn 𝑛 ↔ 𝑓 Fn 1o ) |
8 |
7
|
bicomi |
⊢ ( 𝑓 Fn 1o ↔ [ 1o / 𝑛 ] 𝑓 Fn 𝑛 ) |
9 |
8 2 3
|
3anbi123i |
⊢ ( ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ↔ ( [ 1o / 𝑛 ] 𝑓 Fn 𝑛 ∧ [ 1o / 𝑛 ] 𝜑 ∧ [ 1o / 𝑛 ] 𝜓 ) ) |
10 |
|
sbc3an |
⊢ ( [ 1o / 𝑛 ] ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ↔ ( [ 1o / 𝑛 ] 𝑓 Fn 𝑛 ∧ [ 1o / 𝑛 ] 𝜑 ∧ [ 1o / 𝑛 ] 𝜓 ) ) |
11 |
9 10
|
bitr4i |
⊢ ( ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ↔ [ 1o / 𝑛 ] ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) |
12 |
11
|
eubii |
⊢ ( ∃! 𝑓 ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ↔ ∃! 𝑓 [ 1o / 𝑛 ] ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) |
13 |
6
|
bnj89 |
⊢ ( [ 1o / 𝑛 ] ∃! 𝑓 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ↔ ∃! 𝑓 [ 1o / 𝑛 ] ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) |
14 |
12 13
|
bitr4i |
⊢ ( ∃! 𝑓 ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ↔ [ 1o / 𝑛 ] ∃! 𝑓 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) |
15 |
14
|
imbi2i |
⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ∃! 𝑓 ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → [ 1o / 𝑛 ] ∃! 𝑓 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) ) |
16 |
|
nfv |
⊢ Ⅎ 𝑛 ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) |
17 |
16
|
sbc19.21g |
⊢ ( 1o ∈ V → ( [ 1o / 𝑛 ] ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ∃! 𝑓 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → [ 1o / 𝑛 ] ∃! 𝑓 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) ) ) |
18 |
6 17
|
ax-mp |
⊢ ( [ 1o / 𝑛 ] ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ∃! 𝑓 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → [ 1o / 𝑛 ] ∃! 𝑓 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) ) |
19 |
15 18
|
bitr4i |
⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ∃! 𝑓 ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ) ↔ [ 1o / 𝑛 ] ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ∃! 𝑓 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) ) |
20 |
5 4 19
|
3bitr4i |
⊢ ( 𝜃′ ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ∃! 𝑓 ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ) ) |