Step |
Hyp |
Ref |
Expression |
1 |
|
psrbag.d |
⊢ 𝐷 = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } |
2 |
|
psrbagconf1o.s |
⊢ 𝑆 = { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹 } |
3 |
|
eqid |
⊢ ( 𝑥 ∈ 𝑆 ↦ ( 𝐹 ∘f − 𝑥 ) ) = ( 𝑥 ∈ 𝑆 ↦ ( 𝐹 ∘f − 𝑥 ) ) |
4 |
|
simpll |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ 𝑥 ∈ 𝑆 ) → 𝐼 ∈ 𝑉 ) |
5 |
|
simplr |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ 𝑥 ∈ 𝑆 ) → 𝐹 ∈ 𝐷 ) |
6 |
|
simpr |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ 𝑥 ∈ 𝑆 ) → 𝑥 ∈ 𝑆 ) |
7 |
|
breq1 |
⊢ ( 𝑦 = 𝑥 → ( 𝑦 ∘r ≤ 𝐹 ↔ 𝑥 ∘r ≤ 𝐹 ) ) |
8 |
7 2
|
elrab2 |
⊢ ( 𝑥 ∈ 𝑆 ↔ ( 𝑥 ∈ 𝐷 ∧ 𝑥 ∘r ≤ 𝐹 ) ) |
9 |
6 8
|
sylib |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ 𝑥 ∈ 𝑆 ) → ( 𝑥 ∈ 𝐷 ∧ 𝑥 ∘r ≤ 𝐹 ) ) |
10 |
9
|
simpld |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ 𝑥 ∈ 𝑆 ) → 𝑥 ∈ 𝐷 ) |
11 |
1
|
psrbagfOLD |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑥 ∈ 𝐷 ) → 𝑥 : 𝐼 ⟶ ℕ0 ) |
12 |
4 10 11
|
syl2anc |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ 𝑥 ∈ 𝑆 ) → 𝑥 : 𝐼 ⟶ ℕ0 ) |
13 |
9
|
simprd |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ 𝑥 ∈ 𝑆 ) → 𝑥 ∘r ≤ 𝐹 ) |
14 |
1
|
psrbagconOLD |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∈ 𝐷 ∧ 𝑥 : 𝐼 ⟶ ℕ0 ∧ 𝑥 ∘r ≤ 𝐹 ) ) → ( ( 𝐹 ∘f − 𝑥 ) ∈ 𝐷 ∧ ( 𝐹 ∘f − 𝑥 ) ∘r ≤ 𝐹 ) ) |
15 |
4 5 12 13 14
|
syl13anc |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ 𝑥 ∈ 𝑆 ) → ( ( 𝐹 ∘f − 𝑥 ) ∈ 𝐷 ∧ ( 𝐹 ∘f − 𝑥 ) ∘r ≤ 𝐹 ) ) |
16 |
|
breq1 |
⊢ ( 𝑦 = ( 𝐹 ∘f − 𝑥 ) → ( 𝑦 ∘r ≤ 𝐹 ↔ ( 𝐹 ∘f − 𝑥 ) ∘r ≤ 𝐹 ) ) |
17 |
16 2
|
elrab2 |
⊢ ( ( 𝐹 ∘f − 𝑥 ) ∈ 𝑆 ↔ ( ( 𝐹 ∘f − 𝑥 ) ∈ 𝐷 ∧ ( 𝐹 ∘f − 𝑥 ) ∘r ≤ 𝐹 ) ) |
18 |
15 17
|
sylibr |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ 𝑥 ∈ 𝑆 ) → ( 𝐹 ∘f − 𝑥 ) ∈ 𝑆 ) |
19 |
18
|
ralrimiva |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) → ∀ 𝑥 ∈ 𝑆 ( 𝐹 ∘f − 𝑥 ) ∈ 𝑆 ) |
20 |
|
oveq2 |
⊢ ( 𝑥 = 𝑧 → ( 𝐹 ∘f − 𝑥 ) = ( 𝐹 ∘f − 𝑧 ) ) |
21 |
20
|
eleq1d |
⊢ ( 𝑥 = 𝑧 → ( ( 𝐹 ∘f − 𝑥 ) ∈ 𝑆 ↔ ( 𝐹 ∘f − 𝑧 ) ∈ 𝑆 ) ) |
22 |
21
|
rspccva |
⊢ ( ( ∀ 𝑥 ∈ 𝑆 ( 𝐹 ∘f − 𝑥 ) ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) → ( 𝐹 ∘f − 𝑧 ) ∈ 𝑆 ) |
23 |
19 22
|
sylan |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ 𝑧 ∈ 𝑆 ) → ( 𝐹 ∘f − 𝑧 ) ∈ 𝑆 ) |
24 |
1
|
psrbagfOLD |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) → 𝐹 : 𝐼 ⟶ ℕ0 ) |
25 |
24
|
adantr |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → 𝐹 : 𝐼 ⟶ ℕ0 ) |
26 |
25
|
ffvelrnda |
⊢ ( ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) ∧ 𝑛 ∈ 𝐼 ) → ( 𝐹 ‘ 𝑛 ) ∈ ℕ0 ) |
27 |
|
simpll |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → 𝐼 ∈ 𝑉 ) |
28 |
2
|
ssrab3 |
⊢ 𝑆 ⊆ 𝐷 |
29 |
|
simprr |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → 𝑧 ∈ 𝑆 ) |
30 |
28 29
|
sselid |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → 𝑧 ∈ 𝐷 ) |
31 |
1
|
psrbagfOLD |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝐷 ) → 𝑧 : 𝐼 ⟶ ℕ0 ) |
32 |
27 30 31
|
syl2anc |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → 𝑧 : 𝐼 ⟶ ℕ0 ) |
33 |
32
|
ffvelrnda |
⊢ ( ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) ∧ 𝑛 ∈ 𝐼 ) → ( 𝑧 ‘ 𝑛 ) ∈ ℕ0 ) |
34 |
12
|
adantrr |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → 𝑥 : 𝐼 ⟶ ℕ0 ) |
35 |
34
|
ffvelrnda |
⊢ ( ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) ∧ 𝑛 ∈ 𝐼 ) → ( 𝑥 ‘ 𝑛 ) ∈ ℕ0 ) |
36 |
|
nn0cn |
⊢ ( ( 𝐹 ‘ 𝑛 ) ∈ ℕ0 → ( 𝐹 ‘ 𝑛 ) ∈ ℂ ) |
37 |
|
nn0cn |
⊢ ( ( 𝑧 ‘ 𝑛 ) ∈ ℕ0 → ( 𝑧 ‘ 𝑛 ) ∈ ℂ ) |
38 |
|
nn0cn |
⊢ ( ( 𝑥 ‘ 𝑛 ) ∈ ℕ0 → ( 𝑥 ‘ 𝑛 ) ∈ ℂ ) |
39 |
|
subsub23 |
⊢ ( ( ( 𝐹 ‘ 𝑛 ) ∈ ℂ ∧ ( 𝑧 ‘ 𝑛 ) ∈ ℂ ∧ ( 𝑥 ‘ 𝑛 ) ∈ ℂ ) → ( ( ( 𝐹 ‘ 𝑛 ) − ( 𝑧 ‘ 𝑛 ) ) = ( 𝑥 ‘ 𝑛 ) ↔ ( ( 𝐹 ‘ 𝑛 ) − ( 𝑥 ‘ 𝑛 ) ) = ( 𝑧 ‘ 𝑛 ) ) ) |
40 |
36 37 38 39
|
syl3an |
⊢ ( ( ( 𝐹 ‘ 𝑛 ) ∈ ℕ0 ∧ ( 𝑧 ‘ 𝑛 ) ∈ ℕ0 ∧ ( 𝑥 ‘ 𝑛 ) ∈ ℕ0 ) → ( ( ( 𝐹 ‘ 𝑛 ) − ( 𝑧 ‘ 𝑛 ) ) = ( 𝑥 ‘ 𝑛 ) ↔ ( ( 𝐹 ‘ 𝑛 ) − ( 𝑥 ‘ 𝑛 ) ) = ( 𝑧 ‘ 𝑛 ) ) ) |
41 |
26 33 35 40
|
syl3anc |
⊢ ( ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) ∧ 𝑛 ∈ 𝐼 ) → ( ( ( 𝐹 ‘ 𝑛 ) − ( 𝑧 ‘ 𝑛 ) ) = ( 𝑥 ‘ 𝑛 ) ↔ ( ( 𝐹 ‘ 𝑛 ) − ( 𝑥 ‘ 𝑛 ) ) = ( 𝑧 ‘ 𝑛 ) ) ) |
42 |
|
eqcom |
⊢ ( ( 𝑥 ‘ 𝑛 ) = ( ( 𝐹 ‘ 𝑛 ) − ( 𝑧 ‘ 𝑛 ) ) ↔ ( ( 𝐹 ‘ 𝑛 ) − ( 𝑧 ‘ 𝑛 ) ) = ( 𝑥 ‘ 𝑛 ) ) |
43 |
|
eqcom |
⊢ ( ( 𝑧 ‘ 𝑛 ) = ( ( 𝐹 ‘ 𝑛 ) − ( 𝑥 ‘ 𝑛 ) ) ↔ ( ( 𝐹 ‘ 𝑛 ) − ( 𝑥 ‘ 𝑛 ) ) = ( 𝑧 ‘ 𝑛 ) ) |
44 |
41 42 43
|
3bitr4g |
⊢ ( ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) ∧ 𝑛 ∈ 𝐼 ) → ( ( 𝑥 ‘ 𝑛 ) = ( ( 𝐹 ‘ 𝑛 ) − ( 𝑧 ‘ 𝑛 ) ) ↔ ( 𝑧 ‘ 𝑛 ) = ( ( 𝐹 ‘ 𝑛 ) − ( 𝑥 ‘ 𝑛 ) ) ) ) |
45 |
25
|
ffnd |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → 𝐹 Fn 𝐼 ) |
46 |
32
|
ffnd |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → 𝑧 Fn 𝐼 ) |
47 |
|
inidm |
⊢ ( 𝐼 ∩ 𝐼 ) = 𝐼 |
48 |
|
eqidd |
⊢ ( ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) ∧ 𝑛 ∈ 𝐼 ) → ( 𝐹 ‘ 𝑛 ) = ( 𝐹 ‘ 𝑛 ) ) |
49 |
|
eqidd |
⊢ ( ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) ∧ 𝑛 ∈ 𝐼 ) → ( 𝑧 ‘ 𝑛 ) = ( 𝑧 ‘ 𝑛 ) ) |
50 |
45 46 27 27 47 48 49
|
ofval |
⊢ ( ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) ∧ 𝑛 ∈ 𝐼 ) → ( ( 𝐹 ∘f − 𝑧 ) ‘ 𝑛 ) = ( ( 𝐹 ‘ 𝑛 ) − ( 𝑧 ‘ 𝑛 ) ) ) |
51 |
50
|
eqeq2d |
⊢ ( ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) ∧ 𝑛 ∈ 𝐼 ) → ( ( 𝑥 ‘ 𝑛 ) = ( ( 𝐹 ∘f − 𝑧 ) ‘ 𝑛 ) ↔ ( 𝑥 ‘ 𝑛 ) = ( ( 𝐹 ‘ 𝑛 ) − ( 𝑧 ‘ 𝑛 ) ) ) ) |
52 |
34
|
ffnd |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → 𝑥 Fn 𝐼 ) |
53 |
|
eqidd |
⊢ ( ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) ∧ 𝑛 ∈ 𝐼 ) → ( 𝑥 ‘ 𝑛 ) = ( 𝑥 ‘ 𝑛 ) ) |
54 |
45 52 27 27 47 48 53
|
ofval |
⊢ ( ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) ∧ 𝑛 ∈ 𝐼 ) → ( ( 𝐹 ∘f − 𝑥 ) ‘ 𝑛 ) = ( ( 𝐹 ‘ 𝑛 ) − ( 𝑥 ‘ 𝑛 ) ) ) |
55 |
54
|
eqeq2d |
⊢ ( ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) ∧ 𝑛 ∈ 𝐼 ) → ( ( 𝑧 ‘ 𝑛 ) = ( ( 𝐹 ∘f − 𝑥 ) ‘ 𝑛 ) ↔ ( 𝑧 ‘ 𝑛 ) = ( ( 𝐹 ‘ 𝑛 ) − ( 𝑥 ‘ 𝑛 ) ) ) ) |
56 |
44 51 55
|
3bitr4d |
⊢ ( ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) ∧ 𝑛 ∈ 𝐼 ) → ( ( 𝑥 ‘ 𝑛 ) = ( ( 𝐹 ∘f − 𝑧 ) ‘ 𝑛 ) ↔ ( 𝑧 ‘ 𝑛 ) = ( ( 𝐹 ∘f − 𝑥 ) ‘ 𝑛 ) ) ) |
57 |
56
|
ralbidva |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( ∀ 𝑛 ∈ 𝐼 ( 𝑥 ‘ 𝑛 ) = ( ( 𝐹 ∘f − 𝑧 ) ‘ 𝑛 ) ↔ ∀ 𝑛 ∈ 𝐼 ( 𝑧 ‘ 𝑛 ) = ( ( 𝐹 ∘f − 𝑥 ) ‘ 𝑛 ) ) ) |
58 |
23
|
adantrl |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( 𝐹 ∘f − 𝑧 ) ∈ 𝑆 ) |
59 |
28 58
|
sselid |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( 𝐹 ∘f − 𝑧 ) ∈ 𝐷 ) |
60 |
1
|
psrbagfOLD |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∘f − 𝑧 ) ∈ 𝐷 ) → ( 𝐹 ∘f − 𝑧 ) : 𝐼 ⟶ ℕ0 ) |
61 |
27 59 60
|
syl2anc |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( 𝐹 ∘f − 𝑧 ) : 𝐼 ⟶ ℕ0 ) |
62 |
61
|
ffnd |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( 𝐹 ∘f − 𝑧 ) Fn 𝐼 ) |
63 |
|
eqfnfv |
⊢ ( ( 𝑥 Fn 𝐼 ∧ ( 𝐹 ∘f − 𝑧 ) Fn 𝐼 ) → ( 𝑥 = ( 𝐹 ∘f − 𝑧 ) ↔ ∀ 𝑛 ∈ 𝐼 ( 𝑥 ‘ 𝑛 ) = ( ( 𝐹 ∘f − 𝑧 ) ‘ 𝑛 ) ) ) |
64 |
52 62 63
|
syl2anc |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( 𝑥 = ( 𝐹 ∘f − 𝑧 ) ↔ ∀ 𝑛 ∈ 𝐼 ( 𝑥 ‘ 𝑛 ) = ( ( 𝐹 ∘f − 𝑧 ) ‘ 𝑛 ) ) ) |
65 |
18
|
adantrr |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( 𝐹 ∘f − 𝑥 ) ∈ 𝑆 ) |
66 |
28 65
|
sselid |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( 𝐹 ∘f − 𝑥 ) ∈ 𝐷 ) |
67 |
1
|
psrbagfOLD |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝐹 ∘f − 𝑥 ) ∈ 𝐷 ) → ( 𝐹 ∘f − 𝑥 ) : 𝐼 ⟶ ℕ0 ) |
68 |
27 66 67
|
syl2anc |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( 𝐹 ∘f − 𝑥 ) : 𝐼 ⟶ ℕ0 ) |
69 |
68
|
ffnd |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( 𝐹 ∘f − 𝑥 ) Fn 𝐼 ) |
70 |
|
eqfnfv |
⊢ ( ( 𝑧 Fn 𝐼 ∧ ( 𝐹 ∘f − 𝑥 ) Fn 𝐼 ) → ( 𝑧 = ( 𝐹 ∘f − 𝑥 ) ↔ ∀ 𝑛 ∈ 𝐼 ( 𝑧 ‘ 𝑛 ) = ( ( 𝐹 ∘f − 𝑥 ) ‘ 𝑛 ) ) ) |
71 |
46 69 70
|
syl2anc |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( 𝑧 = ( 𝐹 ∘f − 𝑥 ) ↔ ∀ 𝑛 ∈ 𝐼 ( 𝑧 ‘ 𝑛 ) = ( ( 𝐹 ∘f − 𝑥 ) ‘ 𝑛 ) ) ) |
72 |
57 64 71
|
3bitr4d |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( 𝑥 = ( 𝐹 ∘f − 𝑧 ) ↔ 𝑧 = ( 𝐹 ∘f − 𝑥 ) ) ) |
73 |
3 18 23 72
|
f1o2d |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ) → ( 𝑥 ∈ 𝑆 ↦ ( 𝐹 ∘f − 𝑥 ) ) : 𝑆 –1-1-onto→ 𝑆 ) |