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Metamath Proof Explorer

Theorem ssc2

Description: Infer subset relation on morphisms from the subcategory subset relation. (Contributed by Mario Carneiro, 6-Jan-2017)

Ref Expression
Hypotheses ssc2.1 
|- ( ph -> H Fn ( S X. S ) )
ssc2.2 
|- ( ph -> H C_cat J )
ssc2.3 
|- ( ph -> X e. S )
ssc2.4 
|- ( ph -> Y e. S )
Assertion ssc2 
|- ( ph -> ( X H Y ) C_ ( X J Y ) )

Proof

Step Hyp Ref Expression
1 ssc2.1 
 |-  ( ph -> H Fn ( S X. S ) )
2 ssc2.2 
 |-  ( ph -> H C_cat J )
3 ssc2.3 
 |-  ( ph -> X e. S )
4 ssc2.4 
 |-  ( ph -> Y e. S )
5 eqidd 
 |-  ( ph -> dom dom J = dom dom J )
6 2 5 sscfn2 
 |-  ( ph -> J Fn ( dom dom J X. dom dom J ) )
7 sscrel 
 |-  Rel C_cat
8 7 brrelex2i 
 |-  ( H C_cat J -> J e. _V )
9 dmexg 
 |-  ( J e. _V -> dom J e. _V )
10 dmexg 
 |-  ( dom J e. _V -> dom dom J e. _V )
11 2 8 9 10 4syl 
 |-  ( ph -> dom dom J e. _V )
12 1 6 11 isssc 
 |-  ( ph -> ( H C_cat J <-> ( S C_ dom dom J /\ A. x e. S A. y e. S ( x H y ) C_ ( x J y ) ) ) )
13 2 12 mpbid 
 |-  ( ph -> ( S C_ dom dom J /\ A. x e. S A. y e. S ( x H y ) C_ ( x J y ) ) )
14 13 simprd 
 |-  ( ph -> A. x e. S A. y e. S ( x H y ) C_ ( x J y ) )
15 oveq1 
 |-  ( x = X -> ( x H y ) = ( X H y ) )
16 oveq1 
 |-  ( x = X -> ( x J y ) = ( X J y ) )
17 15 16 sseq12d 
 |-  ( x = X -> ( ( x H y ) C_ ( x J y ) <-> ( X H y ) C_ ( X J y ) ) )
18 oveq2 
 |-  ( y = Y -> ( X H y ) = ( X H Y ) )
19 oveq2 
 |-  ( y = Y -> ( X J y ) = ( X J Y ) )
20 18 19 sseq12d 
 |-  ( y = Y -> ( ( X H y ) C_ ( X J y ) <-> ( X H Y ) C_ ( X J Y ) ) )
21 17 20 rspc2va 
 |-  ( ( ( X e. S /\ Y e. S ) /\ A. x e. S A. y e. S ( x H y ) C_ ( x J y ) ) -> ( X H Y ) C_ ( X J Y ) )
22 3 4 14 21 syl21anc 
 |-  ( ph -> ( X H Y ) C_ ( X J Y ) )