Theorem subcrcl 16476

Description: Reverse closure for the subcategory predicate. (Contributed by Mario Carneiro, 6-Jan-2017.)

Assertion

Ref	Expression
subcrcl	|- ( H e. (Subcat ` C ) -> C e. Cat )

Proof of Theorem subcrcl

Dummy variables f c g h s x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-subc 16472	. . 3 |- Subcat = ( c e. Cat |-> { h | ( h C_cat ( Hom f ` c ) /\ [. dom dom h / s ]. A. x e. s ( ( ( Id ` c ) ` x ) e. ( x h x ) /\ A. y e. s A. z e. s A. f e. ( x h y ) A. g e. ( y h z ) ( g ( <. x , y >. (comp ` c ) z ) f ) e. ( x h z ) ) ) } )
2	1	dmmptss 5631	. 2 |- dom Subcat C_ Cat
3		elfvdm 6220	. 2 |- ( H e. (Subcat ` C ) -> C e. dom Subcat )
4	2, 3	sseldi 3601	1 |- ( H e. (Subcat ` C ) -> C e. Cat )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   {cab 2608   A.wral 2912   [.wsbc 3435   <.cop 4183   class class class wbr 4653   dom cdm 5114   ` cfv 5888  (class class class)co 6650  compcco 15953   Catccat 16325   Idccid 16326   Hom _f chomf 16327   C__cat cssc 16467  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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

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-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fv 5896  df-subc 16472

This theorem is referenced by:  subcssc  16500  subcidcl  16504  subccocl  16505  subccatid  16506  subsubc  16513  funcres2b  16557  funcres2  16558  idfusubc  41866