Theorem brssc 16474

Description: The subcategory subset relation is a relation. (Contributed by Mario Carneiro, 6-Jan-2017.)

Assertion

Ref	Expression
brssc	|- ( H C_cat J <-> E. t ( J Fn ( t X. t ) /\ E. s e. ~P t H e. X_ x e. ( s X. s ) ~P ( J ` x ) ) )

Distinct variable groups:   t, s, x, H   J, s, t, x

Proof of Theorem brssc

Dummy variables h j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sscrel 16473	. . 3 |- Rel C_cat
2		brrelex12 5155	. . 3 |- ( ( Rel C_cat /\ H C_cat J ) -> ( H e. _V /\ J e. _V ) )
3	1, 2	mpan 706	. 2 |- ( H C_cat J -> ( H e. _V /\ J e. _V ) )
4		vex 3203	. . . . . 6 |- t e. _V
5	4, 4	xpex 6962	. . . . 5 |- ( t X. t ) e. _V
6		fnex 6481	. . . . 5 |- ( ( J Fn ( t X. t ) /\ ( t X. t ) e. _V ) -> J e. _V )
7	5, 6	mpan2 707	. . . 4 |- ( J Fn ( t X. t ) -> J e. _V )
8		elex 3212	. . . . 5 |- ( H e. X_ x e. ( s X. s ) ~P ( J ` x ) -> H e. _V )
9	8	rexlimivw 3029	. . . 4 |- ( E. s e. ~P t H e. X_ x e. ( s X. s ) ~P ( J ` x ) -> H e. _V )
10	7, 9	anim12ci 591	. . 3 |- ( ( J Fn ( t X. t ) /\ E. s e. ~P t H e. X_ x e. ( s X. s ) ~P ( J ` x ) ) -> ( H e. _V /\ J e. _V ) )
11	10	exlimiv 1858	. 2 |- ( E. t ( J Fn ( t X. t ) /\ E. s e. ~P t H e. X_ x e. ( s X. s ) ~P ( J ` x ) ) -> ( H e. _V /\ J e. _V ) )
12		simpr 477	. . . . . 6 |- ( ( h = H /\ j = J ) -> j = J )
13	12	fneq1d 5981	. . . . 5 |- ( ( h = H /\ j = J ) -> ( j Fn ( t X. t ) <-> J Fn ( t X. t ) ) )
14		simpl 473	. . . . . . 7 |- ( ( h = H /\ j = J ) -> h = H )
15	12	fveq1d 6193	. . . . . . . . 9 |- ( ( h = H /\ j = J ) -> ( j ` x ) = ( J ` x ) )
16	15	pweqd 4163	. . . . . . . 8 |- ( ( h = H /\ j = J ) -> ~P ( j ` x ) = ~P ( J ` x ) )
17	16	ixpeq2dv 7924	. . . . . . 7 |- ( ( h = H /\ j = J ) -> X_ x e. ( s X. s ) ~P ( j ` x ) = X_ x e. ( s X. s ) ~P ( J ` x ) )
18	14, 17	eleq12d 2695	. . . . . 6 |- ( ( h = H /\ j = J ) -> ( h e. X_ x e. ( s X. s ) ~P ( j ` x ) <-> H e. X_ x e. ( s X. s ) ~P ( J ` x ) ) )
19	18	rexbidv 3052	. . . . 5 |- ( ( h = H /\ j = J ) -> ( E. s e. ~P t h e. X_ x e. ( s X. s ) ~P ( j ` x ) <-> E. s e. ~P t H e. X_ x e. ( s X. s ) ~P ( J ` x ) ) )
20	13, 19	anbi12d 747	. . . 4 |- ( ( h = H /\ j = J ) -> ( ( j Fn ( t X. t ) /\ E. s e. ~P t h e. X_ x e. ( s X. s ) ~P ( j ` x ) ) <-> ( J Fn ( t X. t ) /\ E. s e. ~P t H e. X_ x e. ( s X. s ) ~P ( J ` x ) ) ) )
21	20	exbidv 1850	. . 3 |- ( ( h = H /\ j = J ) -> ( E. t ( j Fn ( t X. t ) /\ E. s e. ~P t h e. X_ x e. ( s X. s ) ~P ( j ` x ) ) <-> E. t ( J Fn ( t X. t ) /\ E. s e. ~P t H e. X_ x e. ( s X. s ) ~P ( J ` x ) ) ) )
22		df-ssc 16470	. . 3 |- C_cat = { <. h , j >. | E. t ( j Fn ( t X. t ) /\ E. s e. ~P t h e. X_ x e. ( s X. s ) ~P ( j ` x ) ) }
23	21, 22	brabga 4989	. 2 |- ( ( H e. _V /\ J e. _V ) -> ( H C_cat J <-> E. t ( J Fn ( t X. t ) /\ E. s e. ~P t H e. X_ x e. ( s X. s ) ~P ( J ` x ) ) ) )
24	3, 11, 23	pm5.21nii 368	1 |- ( H C_cat J <-> E. t ( J Fn ( t X. t ) /\ E. s e. ~P t H e. X_ x e. ( s X. s ) ~P ( J ` x ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   _Vcvv 3200   ~Pcpw 4158   class class class wbr 4653   X. cxp 5112   Rel wrel 5119   Fn wfn 5883   ` cfv 5888   X_cixp 7908   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-ixp 7909  df-ssc 16470

This theorem is referenced by:  sscpwex  16475  sscfn1  16477  sscfn2  16478  isssc  16480