Step |
Hyp |
Ref |
Expression |
1 |
|
ciclcl |
⊢ ( ( 𝐶 ∈ Cat ∧ 𝑅 ( ≃𝑐 ‘ 𝐶 ) 𝑆 ) → 𝑅 ∈ ( Base ‘ 𝐶 ) ) |
2 |
|
cicrcl |
⊢ ( ( 𝐶 ∈ Cat ∧ 𝑅 ( ≃𝑐 ‘ 𝐶 ) 𝑆 ) → 𝑆 ∈ ( Base ‘ 𝐶 ) ) |
3 |
1 2
|
jca |
⊢ ( ( 𝐶 ∈ Cat ∧ 𝑅 ( ≃𝑐 ‘ 𝐶 ) 𝑆 ) → ( 𝑅 ∈ ( Base ‘ 𝐶 ) ∧ 𝑆 ∈ ( Base ‘ 𝐶 ) ) ) |
4 |
3
|
ex |
⊢ ( 𝐶 ∈ Cat → ( 𝑅 ( ≃𝑐 ‘ 𝐶 ) 𝑆 → ( 𝑅 ∈ ( Base ‘ 𝐶 ) ∧ 𝑆 ∈ ( Base ‘ 𝐶 ) ) ) ) |
5 |
|
cicrcl |
⊢ ( ( 𝐶 ∈ Cat ∧ 𝑆 ( ≃𝑐 ‘ 𝐶 ) 𝑇 ) → 𝑇 ∈ ( Base ‘ 𝐶 ) ) |
6 |
5
|
ex |
⊢ ( 𝐶 ∈ Cat → ( 𝑆 ( ≃𝑐 ‘ 𝐶 ) 𝑇 → 𝑇 ∈ ( Base ‘ 𝐶 ) ) ) |
7 |
4 6
|
anim12d |
⊢ ( 𝐶 ∈ Cat → ( ( 𝑅 ( ≃𝑐 ‘ 𝐶 ) 𝑆 ∧ 𝑆 ( ≃𝑐 ‘ 𝐶 ) 𝑇 ) → ( ( 𝑅 ∈ ( Base ‘ 𝐶 ) ∧ 𝑆 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑇 ∈ ( Base ‘ 𝐶 ) ) ) ) |
8 |
7
|
3impib |
⊢ ( ( 𝐶 ∈ Cat ∧ 𝑅 ( ≃𝑐 ‘ 𝐶 ) 𝑆 ∧ 𝑆 ( ≃𝑐 ‘ 𝐶 ) 𝑇 ) → ( ( 𝑅 ∈ ( Base ‘ 𝐶 ) ∧ 𝑆 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑇 ∈ ( Base ‘ 𝐶 ) ) ) |
9 |
|
eqid |
⊢ ( Iso ‘ 𝐶 ) = ( Iso ‘ 𝐶 ) |
10 |
|
eqid |
⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 ) |
11 |
|
simpl |
⊢ ( ( 𝐶 ∈ Cat ∧ ( ( 𝑅 ∈ ( Base ‘ 𝐶 ) ∧ 𝑆 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑇 ∈ ( Base ‘ 𝐶 ) ) ) → 𝐶 ∈ Cat ) |
12 |
|
simpll |
⊢ ( ( ( 𝑅 ∈ ( Base ‘ 𝐶 ) ∧ 𝑆 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑇 ∈ ( Base ‘ 𝐶 ) ) → 𝑅 ∈ ( Base ‘ 𝐶 ) ) |
13 |
12
|
adantl |
⊢ ( ( 𝐶 ∈ Cat ∧ ( ( 𝑅 ∈ ( Base ‘ 𝐶 ) ∧ 𝑆 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑇 ∈ ( Base ‘ 𝐶 ) ) ) → 𝑅 ∈ ( Base ‘ 𝐶 ) ) |
14 |
|
simplr |
⊢ ( ( ( 𝑅 ∈ ( Base ‘ 𝐶 ) ∧ 𝑆 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑇 ∈ ( Base ‘ 𝐶 ) ) → 𝑆 ∈ ( Base ‘ 𝐶 ) ) |
15 |
14
|
adantl |
⊢ ( ( 𝐶 ∈ Cat ∧ ( ( 𝑅 ∈ ( Base ‘ 𝐶 ) ∧ 𝑆 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑇 ∈ ( Base ‘ 𝐶 ) ) ) → 𝑆 ∈ ( Base ‘ 𝐶 ) ) |
16 |
9 10 11 13 15
|
cic |
⊢ ( ( 𝐶 ∈ Cat ∧ ( ( 𝑅 ∈ ( Base ‘ 𝐶 ) ∧ 𝑆 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑇 ∈ ( Base ‘ 𝐶 ) ) ) → ( 𝑅 ( ≃𝑐 ‘ 𝐶 ) 𝑆 ↔ ∃ 𝑓 𝑓 ∈ ( 𝑅 ( Iso ‘ 𝐶 ) 𝑆 ) ) ) |
17 |
|
simprr |
⊢ ( ( 𝐶 ∈ Cat ∧ ( ( 𝑅 ∈ ( Base ‘ 𝐶 ) ∧ 𝑆 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑇 ∈ ( Base ‘ 𝐶 ) ) ) → 𝑇 ∈ ( Base ‘ 𝐶 ) ) |
18 |
9 10 11 15 17
|
cic |
⊢ ( ( 𝐶 ∈ Cat ∧ ( ( 𝑅 ∈ ( Base ‘ 𝐶 ) ∧ 𝑆 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑇 ∈ ( Base ‘ 𝐶 ) ) ) → ( 𝑆 ( ≃𝑐 ‘ 𝐶 ) 𝑇 ↔ ∃ 𝑔 𝑔 ∈ ( 𝑆 ( Iso ‘ 𝐶 ) 𝑇 ) ) ) |
19 |
16 18
|
anbi12d |
⊢ ( ( 𝐶 ∈ Cat ∧ ( ( 𝑅 ∈ ( Base ‘ 𝐶 ) ∧ 𝑆 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑇 ∈ ( Base ‘ 𝐶 ) ) ) → ( ( 𝑅 ( ≃𝑐 ‘ 𝐶 ) 𝑆 ∧ 𝑆 ( ≃𝑐 ‘ 𝐶 ) 𝑇 ) ↔ ( ∃ 𝑓 𝑓 ∈ ( 𝑅 ( Iso ‘ 𝐶 ) 𝑆 ) ∧ ∃ 𝑔 𝑔 ∈ ( 𝑆 ( Iso ‘ 𝐶 ) 𝑇 ) ) ) ) |
20 |
11
|
adantl |
⊢ ( ( ( 𝑔 ∈ ( 𝑆 ( Iso ‘ 𝐶 ) 𝑇 ) ∧ 𝑓 ∈ ( 𝑅 ( Iso ‘ 𝐶 ) 𝑆 ) ) ∧ ( 𝐶 ∈ Cat ∧ ( ( 𝑅 ∈ ( Base ‘ 𝐶 ) ∧ 𝑆 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑇 ∈ ( Base ‘ 𝐶 ) ) ) ) → 𝐶 ∈ Cat ) |
21 |
13
|
adantl |
⊢ ( ( ( 𝑔 ∈ ( 𝑆 ( Iso ‘ 𝐶 ) 𝑇 ) ∧ 𝑓 ∈ ( 𝑅 ( Iso ‘ 𝐶 ) 𝑆 ) ) ∧ ( 𝐶 ∈ Cat ∧ ( ( 𝑅 ∈ ( Base ‘ 𝐶 ) ∧ 𝑆 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑇 ∈ ( Base ‘ 𝐶 ) ) ) ) → 𝑅 ∈ ( Base ‘ 𝐶 ) ) |
22 |
17
|
adantl |
⊢ ( ( ( 𝑔 ∈ ( 𝑆 ( Iso ‘ 𝐶 ) 𝑇 ) ∧ 𝑓 ∈ ( 𝑅 ( Iso ‘ 𝐶 ) 𝑆 ) ) ∧ ( 𝐶 ∈ Cat ∧ ( ( 𝑅 ∈ ( Base ‘ 𝐶 ) ∧ 𝑆 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑇 ∈ ( Base ‘ 𝐶 ) ) ) ) → 𝑇 ∈ ( Base ‘ 𝐶 ) ) |
23 |
|
eqid |
⊢ ( comp ‘ 𝐶 ) = ( comp ‘ 𝐶 ) |
24 |
15
|
adantl |
⊢ ( ( ( 𝑔 ∈ ( 𝑆 ( Iso ‘ 𝐶 ) 𝑇 ) ∧ 𝑓 ∈ ( 𝑅 ( Iso ‘ 𝐶 ) 𝑆 ) ) ∧ ( 𝐶 ∈ Cat ∧ ( ( 𝑅 ∈ ( Base ‘ 𝐶 ) ∧ 𝑆 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑇 ∈ ( Base ‘ 𝐶 ) ) ) ) → 𝑆 ∈ ( Base ‘ 𝐶 ) ) |
25 |
|
simplr |
⊢ ( ( ( 𝑔 ∈ ( 𝑆 ( Iso ‘ 𝐶 ) 𝑇 ) ∧ 𝑓 ∈ ( 𝑅 ( Iso ‘ 𝐶 ) 𝑆 ) ) ∧ ( 𝐶 ∈ Cat ∧ ( ( 𝑅 ∈ ( Base ‘ 𝐶 ) ∧ 𝑆 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑇 ∈ ( Base ‘ 𝐶 ) ) ) ) → 𝑓 ∈ ( 𝑅 ( Iso ‘ 𝐶 ) 𝑆 ) ) |
26 |
|
simpll |
⊢ ( ( ( 𝑔 ∈ ( 𝑆 ( Iso ‘ 𝐶 ) 𝑇 ) ∧ 𝑓 ∈ ( 𝑅 ( Iso ‘ 𝐶 ) 𝑆 ) ) ∧ ( 𝐶 ∈ Cat ∧ ( ( 𝑅 ∈ ( Base ‘ 𝐶 ) ∧ 𝑆 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑇 ∈ ( Base ‘ 𝐶 ) ) ) ) → 𝑔 ∈ ( 𝑆 ( Iso ‘ 𝐶 ) 𝑇 ) ) |
27 |
10 23 9 20 21 24 22 25 26
|
isoco |
⊢ ( ( ( 𝑔 ∈ ( 𝑆 ( Iso ‘ 𝐶 ) 𝑇 ) ∧ 𝑓 ∈ ( 𝑅 ( Iso ‘ 𝐶 ) 𝑆 ) ) ∧ ( 𝐶 ∈ Cat ∧ ( ( 𝑅 ∈ ( Base ‘ 𝐶 ) ∧ 𝑆 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑇 ∈ ( Base ‘ 𝐶 ) ) ) ) → ( 𝑔 ( 〈 𝑅 , 𝑆 〉 ( comp ‘ 𝐶 ) 𝑇 ) 𝑓 ) ∈ ( 𝑅 ( Iso ‘ 𝐶 ) 𝑇 ) ) |
28 |
9 10 20 21 22 27
|
brcici |
⊢ ( ( ( 𝑔 ∈ ( 𝑆 ( Iso ‘ 𝐶 ) 𝑇 ) ∧ 𝑓 ∈ ( 𝑅 ( Iso ‘ 𝐶 ) 𝑆 ) ) ∧ ( 𝐶 ∈ Cat ∧ ( ( 𝑅 ∈ ( Base ‘ 𝐶 ) ∧ 𝑆 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑇 ∈ ( Base ‘ 𝐶 ) ) ) ) → 𝑅 ( ≃𝑐 ‘ 𝐶 ) 𝑇 ) |
29 |
28
|
ex |
⊢ ( ( 𝑔 ∈ ( 𝑆 ( Iso ‘ 𝐶 ) 𝑇 ) ∧ 𝑓 ∈ ( 𝑅 ( Iso ‘ 𝐶 ) 𝑆 ) ) → ( ( 𝐶 ∈ Cat ∧ ( ( 𝑅 ∈ ( Base ‘ 𝐶 ) ∧ 𝑆 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑇 ∈ ( Base ‘ 𝐶 ) ) ) → 𝑅 ( ≃𝑐 ‘ 𝐶 ) 𝑇 ) ) |
30 |
29
|
ex |
⊢ ( 𝑔 ∈ ( 𝑆 ( Iso ‘ 𝐶 ) 𝑇 ) → ( 𝑓 ∈ ( 𝑅 ( Iso ‘ 𝐶 ) 𝑆 ) → ( ( 𝐶 ∈ Cat ∧ ( ( 𝑅 ∈ ( Base ‘ 𝐶 ) ∧ 𝑆 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑇 ∈ ( Base ‘ 𝐶 ) ) ) → 𝑅 ( ≃𝑐 ‘ 𝐶 ) 𝑇 ) ) ) |
31 |
30
|
exlimiv |
⊢ ( ∃ 𝑔 𝑔 ∈ ( 𝑆 ( Iso ‘ 𝐶 ) 𝑇 ) → ( 𝑓 ∈ ( 𝑅 ( Iso ‘ 𝐶 ) 𝑆 ) → ( ( 𝐶 ∈ Cat ∧ ( ( 𝑅 ∈ ( Base ‘ 𝐶 ) ∧ 𝑆 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑇 ∈ ( Base ‘ 𝐶 ) ) ) → 𝑅 ( ≃𝑐 ‘ 𝐶 ) 𝑇 ) ) ) |
32 |
31
|
com12 |
⊢ ( 𝑓 ∈ ( 𝑅 ( Iso ‘ 𝐶 ) 𝑆 ) → ( ∃ 𝑔 𝑔 ∈ ( 𝑆 ( Iso ‘ 𝐶 ) 𝑇 ) → ( ( 𝐶 ∈ Cat ∧ ( ( 𝑅 ∈ ( Base ‘ 𝐶 ) ∧ 𝑆 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑇 ∈ ( Base ‘ 𝐶 ) ) ) → 𝑅 ( ≃𝑐 ‘ 𝐶 ) 𝑇 ) ) ) |
33 |
32
|
exlimiv |
⊢ ( ∃ 𝑓 𝑓 ∈ ( 𝑅 ( Iso ‘ 𝐶 ) 𝑆 ) → ( ∃ 𝑔 𝑔 ∈ ( 𝑆 ( Iso ‘ 𝐶 ) 𝑇 ) → ( ( 𝐶 ∈ Cat ∧ ( ( 𝑅 ∈ ( Base ‘ 𝐶 ) ∧ 𝑆 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑇 ∈ ( Base ‘ 𝐶 ) ) ) → 𝑅 ( ≃𝑐 ‘ 𝐶 ) 𝑇 ) ) ) |
34 |
33
|
imp |
⊢ ( ( ∃ 𝑓 𝑓 ∈ ( 𝑅 ( Iso ‘ 𝐶 ) 𝑆 ) ∧ ∃ 𝑔 𝑔 ∈ ( 𝑆 ( Iso ‘ 𝐶 ) 𝑇 ) ) → ( ( 𝐶 ∈ Cat ∧ ( ( 𝑅 ∈ ( Base ‘ 𝐶 ) ∧ 𝑆 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑇 ∈ ( Base ‘ 𝐶 ) ) ) → 𝑅 ( ≃𝑐 ‘ 𝐶 ) 𝑇 ) ) |
35 |
34
|
com12 |
⊢ ( ( 𝐶 ∈ Cat ∧ ( ( 𝑅 ∈ ( Base ‘ 𝐶 ) ∧ 𝑆 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑇 ∈ ( Base ‘ 𝐶 ) ) ) → ( ( ∃ 𝑓 𝑓 ∈ ( 𝑅 ( Iso ‘ 𝐶 ) 𝑆 ) ∧ ∃ 𝑔 𝑔 ∈ ( 𝑆 ( Iso ‘ 𝐶 ) 𝑇 ) ) → 𝑅 ( ≃𝑐 ‘ 𝐶 ) 𝑇 ) ) |
36 |
19 35
|
sylbid |
⊢ ( ( 𝐶 ∈ Cat ∧ ( ( 𝑅 ∈ ( Base ‘ 𝐶 ) ∧ 𝑆 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑇 ∈ ( Base ‘ 𝐶 ) ) ) → ( ( 𝑅 ( ≃𝑐 ‘ 𝐶 ) 𝑆 ∧ 𝑆 ( ≃𝑐 ‘ 𝐶 ) 𝑇 ) → 𝑅 ( ≃𝑐 ‘ 𝐶 ) 𝑇 ) ) |
37 |
36
|
ex |
⊢ ( 𝐶 ∈ Cat → ( ( ( 𝑅 ∈ ( Base ‘ 𝐶 ) ∧ 𝑆 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑇 ∈ ( Base ‘ 𝐶 ) ) → ( ( 𝑅 ( ≃𝑐 ‘ 𝐶 ) 𝑆 ∧ 𝑆 ( ≃𝑐 ‘ 𝐶 ) 𝑇 ) → 𝑅 ( ≃𝑐 ‘ 𝐶 ) 𝑇 ) ) ) |
38 |
37
|
com23 |
⊢ ( 𝐶 ∈ Cat → ( ( 𝑅 ( ≃𝑐 ‘ 𝐶 ) 𝑆 ∧ 𝑆 ( ≃𝑐 ‘ 𝐶 ) 𝑇 ) → ( ( ( 𝑅 ∈ ( Base ‘ 𝐶 ) ∧ 𝑆 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑇 ∈ ( Base ‘ 𝐶 ) ) → 𝑅 ( ≃𝑐 ‘ 𝐶 ) 𝑇 ) ) ) |
39 |
38
|
3impib |
⊢ ( ( 𝐶 ∈ Cat ∧ 𝑅 ( ≃𝑐 ‘ 𝐶 ) 𝑆 ∧ 𝑆 ( ≃𝑐 ‘ 𝐶 ) 𝑇 ) → ( ( ( 𝑅 ∈ ( Base ‘ 𝐶 ) ∧ 𝑆 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑇 ∈ ( Base ‘ 𝐶 ) ) → 𝑅 ( ≃𝑐 ‘ 𝐶 ) 𝑇 ) ) |
40 |
8 39
|
mpd |
⊢ ( ( 𝐶 ∈ Cat ∧ 𝑅 ( ≃𝑐 ‘ 𝐶 ) 𝑆 ∧ 𝑆 ( ≃𝑐 ‘ 𝐶 ) 𝑇 ) → 𝑅 ( ≃𝑐 ‘ 𝐶 ) 𝑇 ) |