Metamath Proof Explorer

Theorem cicpropdlem

Description: Lemma for cicpropd . (Contributed by Zhi Wang, 27-Oct-2025)

Ref Expression
Hypotheses cicpropd.1 
|- ( ph -> ( Homf ` C ) = ( Homf ` D ) )
cicpropd.2 
|- ( ph -> ( comf ` C ) = ( comf ` D ) )
Assertion cicpropdlem 
|- ( ( ph /\ P e. ( ~=c ` C ) ) -> P e. ( ~=c ` D ) )

Proof

Step Hyp Ref Expression
1 cicpropd.1 
 |-  ( ph -> ( Homf ` C ) = ( Homf ` D ) )
2 cicpropd.2 
 |-  ( ph -> ( comf ` C ) = ( comf ` D ) )
3 cic1st2nd 
 |-  ( P e. ( ~=c ` C ) -> P = <. ( 1st ` P ) , ( 2nd ` P ) >. )
4 3 adantl 
 |-  ( ( ph /\ P e. ( ~=c ` C ) ) -> P = <. ( 1st ` P ) , ( 2nd ` P ) >. )
5 cic1st2ndbr 
 |-  ( P e. ( ~=c ` C ) -> ( 1st ` P ) ( ~=c ` C ) ( 2nd ` P ) )
6 5 adantl 
 |-  ( ( ph /\ P e. ( ~=c ` C ) ) -> ( 1st ` P ) ( ~=c ` C ) ( 2nd ` P ) )
7 1 adantr 
 |-  ( ( ph /\ P e. ( ~=c ` C ) ) -> ( Homf ` C ) = ( Homf ` D ) )
8 2 adantr 
 |-  ( ( ph /\ P e. ( ~=c ` C ) ) -> ( comf ` C ) = ( comf ` D ) )
9 7 8 isopropd 
 |-  ( ( ph /\ P e. ( ~=c ` C ) ) -> ( Iso ` C ) = ( Iso ` D ) )
10 9 oveqd 
 |-  ( ( ph /\ P e. ( ~=c ` C ) ) -> ( ( 1st ` P ) ( Iso ` C ) ( 2nd ` P ) ) = ( ( 1st ` P ) ( Iso ` D ) ( 2nd ` P ) ) )
11 10 neeq1d 
 |-  ( ( ph /\ P e. ( ~=c ` C ) ) -> ( ( ( 1st ` P ) ( Iso ` C ) ( 2nd ` P ) ) =/= (/) <-> ( ( 1st ` P ) ( Iso ` D ) ( 2nd ` P ) ) =/= (/) ) )
12 eqid 
 |-  ( Iso ` C ) = ( Iso ` C )
13 eqid 
 |-  ( Base ` C ) = ( Base ` C )
14 cicrcl2 
 |-  ( ( 1st ` P ) ( ~=c ` C ) ( 2nd ` P ) -> C e. Cat )
15 5 14 syl 
 |-  ( P e. ( ~=c ` C ) -> C e. Cat )
16 15 adantl 
 |-  ( ( ph /\ P e. ( ~=c ` C ) ) -> C e. Cat )
17 ciclcl 
 |-  ( ( C e. Cat /\ ( 1st ` P ) ( ~=c ` C ) ( 2nd ` P ) ) -> ( 1st ` P ) e. ( Base ` C ) )
18 14 17 mpancom 
 |-  ( ( 1st ` P ) ( ~=c ` C ) ( 2nd ` P ) -> ( 1st ` P ) e. ( Base ` C ) )
19 5 18 syl 
 |-  ( P e. ( ~=c ` C ) -> ( 1st ` P ) e. ( Base ` C ) )
20 19 adantl 
 |-  ( ( ph /\ P e. ( ~=c ` C ) ) -> ( 1st ` P ) e. ( Base ` C ) )
21 cicrcl 
 |-  ( ( C e. Cat /\ ( 1st ` P ) ( ~=c ` C ) ( 2nd ` P ) ) -> ( 2nd ` P ) e. ( Base ` C ) )
22 14 21 mpancom 
 |-  ( ( 1st ` P ) ( ~=c ` C ) ( 2nd ` P ) -> ( 2nd ` P ) e. ( Base ` C ) )
23 5 22 syl 
 |-  ( P e. ( ~=c ` C ) -> ( 2nd ` P ) e. ( Base ` C ) )
24 23 adantl 
 |-  ( ( ph /\ P e. ( ~=c ` C ) ) -> ( 2nd ` P ) e. ( Base ` C ) )
25 12 13 16 20 24 brcic 
 |-  ( ( ph /\ P e. ( ~=c ` C ) ) -> ( ( 1st ` P ) ( ~=c ` C ) ( 2nd ` P ) <-> ( ( 1st ` P ) ( Iso ` C ) ( 2nd ` P ) ) =/= (/) ) )
26 eqid 
 |-  ( Iso ` D ) = ( Iso ` D )
27 eqid 
 |-  ( Base ` D ) = ( Base ` D )
28 1 homfeqbas 
 |-  ( ph -> ( Base ` C ) = ( Base ` D ) )
29 28 adantr 
 |-  ( ( ph /\ P e. ( ~=c ` C ) ) -> ( Base ` C ) = ( Base ` D ) )
30 20 29 eleqtrd 
 |-  ( ( ph /\ P e. ( ~=c ` C ) ) -> ( 1st ` P ) e. ( Base ` D ) )
31 30 elfvexd 
 |-  ( ( ph /\ P e. ( ~=c ` C ) ) -> D e. _V )
32 7 8 16 31 catpropd 
 |-  ( ( ph /\ P e. ( ~=c ` C ) ) -> ( C e. Cat <-> D e. Cat ) )
33 16 32 mpbid 
 |-  ( ( ph /\ P e. ( ~=c ` C ) ) -> D e. Cat )
34 24 29 eleqtrd 
 |-  ( ( ph /\ P e. ( ~=c ` C ) ) -> ( 2nd ` P ) e. ( Base ` D ) )
35 26 27 33 30 34 brcic 
 |-  ( ( ph /\ P e. ( ~=c ` C ) ) -> ( ( 1st ` P ) ( ~=c ` D ) ( 2nd ` P ) <-> ( ( 1st ` P ) ( Iso ` D ) ( 2nd ` P ) ) =/= (/) ) )
36 11 25 35 3bitr4d 
 |-  ( ( ph /\ P e. ( ~=c ` C ) ) -> ( ( 1st ` P ) ( ~=c ` C ) ( 2nd ` P ) <-> ( 1st ` P ) ( ~=c ` D ) ( 2nd ` P ) ) )
37 6 36 mpbid 
 |-  ( ( ph /\ P e. ( ~=c ` C ) ) -> ( 1st ` P ) ( ~=c ` D ) ( 2nd ` P ) )
38 df-br 
 |-  ( ( 1st ` P ) ( ~=c ` D ) ( 2nd ` P ) <-> <. ( 1st ` P ) , ( 2nd ` P ) >. e. ( ~=c ` D ) )
39 37 38 sylib 
 |-  ( ( ph /\ P e. ( ~=c ` C ) ) -> <. ( 1st ` P ) , ( 2nd ` P ) >. e. ( ~=c ` D ) )
40 4 39 eqeltrd 
 |-  ( ( ph /\ P e. ( ~=c ` C ) ) -> P e. ( ~=c ` D ) )