Theorem ssc1 16481

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

Hypotheses

Ref	Expression
isssc.1	|- ( ph -> H Fn ( S X. S ) )
isssc.2	|- ( ph -> J Fn ( T X. T ) )
ssc1.3	|- ( ph -> H C_cat J )

Assertion

Ref	Expression
ssc1	|- ( ph -> S C_ T )

Proof of Theorem ssc1

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssc1.3	. . 3 |- ( ph -> H C_cat J )
2		isssc.1	. . . 4 |- ( ph -> H Fn ( S X. S ) )
3		isssc.2	. . . 4 |- ( ph -> J Fn ( T X. T ) )
4		sscrel 16473	. . . . . . 7 |- Rel C_cat
5	4	brrelex2i 5159	. . . . . 6 |- ( H C_cat J -> J e. _V )
6	1, 5	syl 17	. . . . 5 |- ( ph -> J e. _V )
7	3	ssclem 16479	. . . . 5 |- ( ph -> ( J e. _V <-> T e. _V ) )
8	6, 7	mpbid 222	. . . 4 |- ( ph -> T e. _V )
9	2, 3, 8	isssc 16480	. . 3 |- ( ph -> ( H C_cat J <-> ( S C_ T /\ A. x e. S A. y e. S ( x H y ) C_ ( x J y ) ) ) )
10	1, 9	mpbid 222	. 2 |- ( ph -> ( S C_ T /\ A. x e. S A. y e. S ( x H y ) C_ ( x J y ) ) )
11	10	simpld 475	1 |- ( ph -> S C_ T )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   X. cxp 5112   Fn wfn 5883  (class class class)co 6650   C__cat cssc 16467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-ixp 7909  df-ssc 16470

This theorem is referenced by:  ssctr  16485  ssceq  16486  subcss1  16502  issubc3  16509  subsubc  16513