Step |
Hyp |
Ref |
Expression |
1 |
|
fmptco1f1o.a |
⊢ 𝐴 = ( 𝑅 ↑m 𝐸 ) |
2 |
|
fmptco1f1o.b |
⊢ 𝐵 = ( 𝑅 ↑m 𝐷 ) |
3 |
|
fmptco1f1o.f |
⊢ 𝐹 = ( 𝑓 ∈ 𝐴 ↦ ( 𝑓 ∘ 𝑇 ) ) |
4 |
|
fmptco1f1o.d |
⊢ ( 𝜑 → 𝐷 ∈ 𝑉 ) |
5 |
|
fmptco1f1o.e |
⊢ ( 𝜑 → 𝐸 ∈ 𝑊 ) |
6 |
|
fmptco1f1o.r |
⊢ ( 𝜑 → 𝑅 ∈ 𝑋 ) |
7 |
|
fmptco1f1o.t |
⊢ ( 𝜑 → 𝑇 : 𝐷 –1-1-onto→ 𝐸 ) |
8 |
3
|
a1i |
⊢ ( 𝜑 → 𝐹 = ( 𝑓 ∈ 𝐴 ↦ ( 𝑓 ∘ 𝑇 ) ) ) |
9 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ) → 𝑅 ∈ 𝑋 ) |
10 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ) → 𝐷 ∈ 𝑉 ) |
11 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ) → 𝑓 ∈ 𝐴 ) |
12 |
11 1
|
eleqtrdi |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ) → 𝑓 ∈ ( 𝑅 ↑m 𝐸 ) ) |
13 |
|
elmapi |
⊢ ( 𝑓 ∈ ( 𝑅 ↑m 𝐸 ) → 𝑓 : 𝐸 ⟶ 𝑅 ) |
14 |
12 13
|
syl |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ) → 𝑓 : 𝐸 ⟶ 𝑅 ) |
15 |
|
f1of |
⊢ ( 𝑇 : 𝐷 –1-1-onto→ 𝐸 → 𝑇 : 𝐷 ⟶ 𝐸 ) |
16 |
7 15
|
syl |
⊢ ( 𝜑 → 𝑇 : 𝐷 ⟶ 𝐸 ) |
17 |
16
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ) → 𝑇 : 𝐷 ⟶ 𝐸 ) |
18 |
|
fco |
⊢ ( ( 𝑓 : 𝐸 ⟶ 𝑅 ∧ 𝑇 : 𝐷 ⟶ 𝐸 ) → ( 𝑓 ∘ 𝑇 ) : 𝐷 ⟶ 𝑅 ) |
19 |
14 17 18
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ) → ( 𝑓 ∘ 𝑇 ) : 𝐷 ⟶ 𝑅 ) |
20 |
|
elmapg |
⊢ ( ( 𝑅 ∈ 𝑋 ∧ 𝐷 ∈ 𝑉 ) → ( ( 𝑓 ∘ 𝑇 ) ∈ ( 𝑅 ↑m 𝐷 ) ↔ ( 𝑓 ∘ 𝑇 ) : 𝐷 ⟶ 𝑅 ) ) |
21 |
20
|
biimpar |
⊢ ( ( ( 𝑅 ∈ 𝑋 ∧ 𝐷 ∈ 𝑉 ) ∧ ( 𝑓 ∘ 𝑇 ) : 𝐷 ⟶ 𝑅 ) → ( 𝑓 ∘ 𝑇 ) ∈ ( 𝑅 ↑m 𝐷 ) ) |
22 |
9 10 19 21
|
syl21anc |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ) → ( 𝑓 ∘ 𝑇 ) ∈ ( 𝑅 ↑m 𝐷 ) ) |
23 |
22 2
|
eleqtrrdi |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ) → ( 𝑓 ∘ 𝑇 ) ∈ 𝐵 ) |
24 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐵 ) → 𝑅 ∈ 𝑋 ) |
25 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐵 ) → 𝐸 ∈ 𝑊 ) |
26 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐵 ) → 𝑔 ∈ 𝐵 ) |
27 |
26 2
|
eleqtrdi |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐵 ) → 𝑔 ∈ ( 𝑅 ↑m 𝐷 ) ) |
28 |
|
elmapi |
⊢ ( 𝑔 ∈ ( 𝑅 ↑m 𝐷 ) → 𝑔 : 𝐷 ⟶ 𝑅 ) |
29 |
27 28
|
syl |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐵 ) → 𝑔 : 𝐷 ⟶ 𝑅 ) |
30 |
|
f1ocnv |
⊢ ( 𝑇 : 𝐷 –1-1-onto→ 𝐸 → ◡ 𝑇 : 𝐸 –1-1-onto→ 𝐷 ) |
31 |
|
f1of |
⊢ ( ◡ 𝑇 : 𝐸 –1-1-onto→ 𝐷 → ◡ 𝑇 : 𝐸 ⟶ 𝐷 ) |
32 |
7 30 31
|
3syl |
⊢ ( 𝜑 → ◡ 𝑇 : 𝐸 ⟶ 𝐷 ) |
33 |
32
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐵 ) → ◡ 𝑇 : 𝐸 ⟶ 𝐷 ) |
34 |
|
fco |
⊢ ( ( 𝑔 : 𝐷 ⟶ 𝑅 ∧ ◡ 𝑇 : 𝐸 ⟶ 𝐷 ) → ( 𝑔 ∘ ◡ 𝑇 ) : 𝐸 ⟶ 𝑅 ) |
35 |
29 33 34
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐵 ) → ( 𝑔 ∘ ◡ 𝑇 ) : 𝐸 ⟶ 𝑅 ) |
36 |
|
elmapg |
⊢ ( ( 𝑅 ∈ 𝑋 ∧ 𝐸 ∈ 𝑊 ) → ( ( 𝑔 ∘ ◡ 𝑇 ) ∈ ( 𝑅 ↑m 𝐸 ) ↔ ( 𝑔 ∘ ◡ 𝑇 ) : 𝐸 ⟶ 𝑅 ) ) |
37 |
36
|
biimpar |
⊢ ( ( ( 𝑅 ∈ 𝑋 ∧ 𝐸 ∈ 𝑊 ) ∧ ( 𝑔 ∘ ◡ 𝑇 ) : 𝐸 ⟶ 𝑅 ) → ( 𝑔 ∘ ◡ 𝑇 ) ∈ ( 𝑅 ↑m 𝐸 ) ) |
38 |
24 25 35 37
|
syl21anc |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐵 ) → ( 𝑔 ∘ ◡ 𝑇 ) ∈ ( 𝑅 ↑m 𝐸 ) ) |
39 |
38 1
|
eleqtrrdi |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐵 ) → ( 𝑔 ∘ ◡ 𝑇 ) ∈ 𝐴 ) |
40 |
|
coass |
⊢ ( ( 𝑔 ∘ ◡ 𝑇 ) ∘ 𝑇 ) = ( 𝑔 ∘ ( ◡ 𝑇 ∘ 𝑇 ) ) |
41 |
7
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ) ∧ 𝑔 ∈ 𝐵 ) → 𝑇 : 𝐷 –1-1-onto→ 𝐸 ) |
42 |
|
f1ococnv1 |
⊢ ( 𝑇 : 𝐷 –1-1-onto→ 𝐸 → ( ◡ 𝑇 ∘ 𝑇 ) = ( I ↾ 𝐷 ) ) |
43 |
42
|
coeq2d |
⊢ ( 𝑇 : 𝐷 –1-1-onto→ 𝐸 → ( 𝑔 ∘ ( ◡ 𝑇 ∘ 𝑇 ) ) = ( 𝑔 ∘ ( I ↾ 𝐷 ) ) ) |
44 |
41 43
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ) ∧ 𝑔 ∈ 𝐵 ) → ( 𝑔 ∘ ( ◡ 𝑇 ∘ 𝑇 ) ) = ( 𝑔 ∘ ( I ↾ 𝐷 ) ) ) |
45 |
29
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ) ∧ 𝑔 ∈ 𝐵 ) → 𝑔 : 𝐷 ⟶ 𝑅 ) |
46 |
|
fcoi1 |
⊢ ( 𝑔 : 𝐷 ⟶ 𝑅 → ( 𝑔 ∘ ( I ↾ 𝐷 ) ) = 𝑔 ) |
47 |
45 46
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ) ∧ 𝑔 ∈ 𝐵 ) → ( 𝑔 ∘ ( I ↾ 𝐷 ) ) = 𝑔 ) |
48 |
44 47
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ) ∧ 𝑔 ∈ 𝐵 ) → ( 𝑔 ∘ ( ◡ 𝑇 ∘ 𝑇 ) ) = 𝑔 ) |
49 |
40 48
|
eqtr2id |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ) ∧ 𝑔 ∈ 𝐵 ) → 𝑔 = ( ( 𝑔 ∘ ◡ 𝑇 ) ∘ 𝑇 ) ) |
50 |
49
|
eqeq1d |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ) ∧ 𝑔 ∈ 𝐵 ) → ( 𝑔 = ( 𝑓 ∘ 𝑇 ) ↔ ( ( 𝑔 ∘ ◡ 𝑇 ) ∘ 𝑇 ) = ( 𝑓 ∘ 𝑇 ) ) ) |
51 |
|
eqcom |
⊢ ( ( ( 𝑔 ∘ ◡ 𝑇 ) ∘ 𝑇 ) = ( 𝑓 ∘ 𝑇 ) ↔ ( 𝑓 ∘ 𝑇 ) = ( ( 𝑔 ∘ ◡ 𝑇 ) ∘ 𝑇 ) ) |
52 |
51
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ) ∧ 𝑔 ∈ 𝐵 ) → ( ( ( 𝑔 ∘ ◡ 𝑇 ) ∘ 𝑇 ) = ( 𝑓 ∘ 𝑇 ) ↔ ( 𝑓 ∘ 𝑇 ) = ( ( 𝑔 ∘ ◡ 𝑇 ) ∘ 𝑇 ) ) ) |
53 |
|
f1ofo |
⊢ ( 𝑇 : 𝐷 –1-1-onto→ 𝐸 → 𝑇 : 𝐷 –onto→ 𝐸 ) |
54 |
41 53
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ) ∧ 𝑔 ∈ 𝐵 ) → 𝑇 : 𝐷 –onto→ 𝐸 ) |
55 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ) ∧ 𝑔 ∈ 𝐵 ) → 𝑓 ∈ 𝐴 ) |
56 |
55 1
|
eleqtrdi |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ) ∧ 𝑔 ∈ 𝐵 ) → 𝑓 ∈ ( 𝑅 ↑m 𝐸 ) ) |
57 |
|
elmapfn |
⊢ ( 𝑓 ∈ ( 𝑅 ↑m 𝐸 ) → 𝑓 Fn 𝐸 ) |
58 |
56 57
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ) ∧ 𝑔 ∈ 𝐵 ) → 𝑓 Fn 𝐸 ) |
59 |
|
elmapfn |
⊢ ( ( 𝑔 ∘ ◡ 𝑇 ) ∈ ( 𝑅 ↑m 𝐸 ) → ( 𝑔 ∘ ◡ 𝑇 ) Fn 𝐸 ) |
60 |
38 59
|
syl |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐵 ) → ( 𝑔 ∘ ◡ 𝑇 ) Fn 𝐸 ) |
61 |
60
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ) ∧ 𝑔 ∈ 𝐵 ) → ( 𝑔 ∘ ◡ 𝑇 ) Fn 𝐸 ) |
62 |
|
cocan2 |
⊢ ( ( 𝑇 : 𝐷 –onto→ 𝐸 ∧ 𝑓 Fn 𝐸 ∧ ( 𝑔 ∘ ◡ 𝑇 ) Fn 𝐸 ) → ( ( 𝑓 ∘ 𝑇 ) = ( ( 𝑔 ∘ ◡ 𝑇 ) ∘ 𝑇 ) ↔ 𝑓 = ( 𝑔 ∘ ◡ 𝑇 ) ) ) |
63 |
54 58 61 62
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ) ∧ 𝑔 ∈ 𝐵 ) → ( ( 𝑓 ∘ 𝑇 ) = ( ( 𝑔 ∘ ◡ 𝑇 ) ∘ 𝑇 ) ↔ 𝑓 = ( 𝑔 ∘ ◡ 𝑇 ) ) ) |
64 |
50 52 63
|
3bitrrd |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ 𝐴 ) ∧ 𝑔 ∈ 𝐵 ) → ( 𝑓 = ( 𝑔 ∘ ◡ 𝑇 ) ↔ 𝑔 = ( 𝑓 ∘ 𝑇 ) ) ) |
65 |
64
|
anasss |
⊢ ( ( 𝜑 ∧ ( 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐵 ) ) → ( 𝑓 = ( 𝑔 ∘ ◡ 𝑇 ) ↔ 𝑔 = ( 𝑓 ∘ 𝑇 ) ) ) |
66 |
8 23 39 65
|
f1o3d |
⊢ ( 𝜑 → ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ ◡ 𝐹 = ( 𝑔 ∈ 𝐵 ↦ ( 𝑔 ∘ ◡ 𝑇 ) ) ) ) |
67 |
66
|
simpld |
⊢ ( 𝜑 → 𝐹 : 𝐴 –1-1-onto→ 𝐵 ) |