Theorem subcss2 16503

Description: The morphisms of a subcategory are a subset of the morphisms of the original. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
subcss1.1	|- ( ph -> J e. (Subcat ` C ) )
subcss1.2	|- ( ph -> J Fn ( S X. S ) )
subcss2.h	|- H = ( Hom ` C )
subcss2.x	|- ( ph -> X e. S )
subcss2.y	|- ( ph -> Y e. S )

Assertion

Ref	Expression
subcss2	|- ( ph -> ( X J Y ) C_ ( X H Y ) )

Proof of Theorem subcss2

Step	Hyp	Ref	Expression
1		subcss1.2	. . 3 |- ( ph -> J Fn ( S X. S ) )
2		subcss1.1	. . . 4 |- ( ph -> J e. (Subcat ` C ) )
3		eqid 2622	. . . 4 |- ( Hom f ` C ) = ( Hom f ` C )
4	2, 3	subcssc 16500	. . 3 |- ( ph -> J C_cat ( Hom f ` C ) )
5		subcss2.x	. . 3 |- ( ph -> X e. S )
6		subcss2.y	. . 3 |- ( ph -> Y e. S )
7	1, 4, 5, 6	ssc2 16482	. 2 |- ( ph -> ( X J Y ) C_ ( X ( Hom f ` C ) Y ) )
8		eqid 2622	. . 3 |- ( Base ` C ) = ( Base ` C )
9		subcss2.h	. . 3 |- H = ( Hom ` C )
10	2, 1, 8	subcss1 16502	. . . 4 |- ( ph -> S C_ ( Base ` C ) )
11	10, 5	sseldd 3604	. . 3 |- ( ph -> X e. ( Base ` C ) )
12	10, 6	sseldd 3604	. . 3 |- ( ph -> Y e. ( Base ` C ) )
13	3, 8, 9, 11, 12	homfval 16352	. 2 |- ( ph -> ( X ( Hom f ` C ) Y ) = ( X H Y ) )
14	7, 13	sseqtrd 3641	1 |- ( ph -> ( X J Y ) C_ ( X H Y ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   C_ wss 3574   X. cxp 5112   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   Basecbs 15857   Hom chom 15952   Hom _f chomf 16327  Subcatcsubc 16469

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-fal 1489  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-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-pm 7860  df-ixp 7909  df-homf 16331  df-ssc 16470  df-subc 16472

This theorem is referenced by:  subccatid  16506  funcres  16556  funcres2b  16557