Theorem rescval 16487

Description: Value of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypothesis

Ref	Expression
rescval.1	|- D = ( C |`cat H )

Assertion

Ref	Expression
rescval	|- ( ( C e. V /\ H e. W ) -> D = ( ( C ↾s dom dom H ) sSet <. ( Hom ` ndx ) , H >. ) )

Proof of Theorem rescval

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

Step	Hyp	Ref	Expression
1		rescval.1	. 2 |- D = ( C |`cat H )
2		elex 3212	. . 3 |- ( C e. V -> C e. _V )
3		elex 3212	. . 3 |- ( H e. W -> H e. _V )
4		simpl 473	. . . . . 6 |- ( ( c = C /\ h = H ) -> c = C )
5		simpr 477	. . . . . . . 8 |- ( ( c = C /\ h = H ) -> h = H )
6	5	dmeqd 5326	. . . . . . 7 |- ( ( c = C /\ h = H ) -> dom h = dom H )
7	6	dmeqd 5326	. . . . . 6 |- ( ( c = C /\ h = H ) -> dom dom h = dom dom H )
8	4, 7	oveq12d 6668	. . . . 5 |- ( ( c = C /\ h = H ) -> ( c ↾s dom dom h ) = ( C ↾s dom dom H ) )
9	5	opeq2d 4409	. . . . 5 |- ( ( c = C /\ h = H ) -> <. ( Hom ` ndx ) , h >. = <. ( Hom ` ndx ) , H >. )
10	8, 9	oveq12d 6668	. . . 4 |- ( ( c = C /\ h = H ) -> ( ( c ↾s dom dom h ) sSet <. ( Hom ` ndx ) , h >. ) = ( ( C ↾s dom dom H ) sSet <. ( Hom ` ndx ) , H >. ) )
11		df-resc 16471	. . . 4 |- |`cat = ( c e. _V , h e. _V |-> ( ( c ↾s dom dom h ) sSet <. ( Hom ` ndx ) , h >. ) )
12		ovex 6678	. . . 4 |- ( ( C ↾s dom dom H ) sSet <. ( Hom ` ndx ) , H >. ) e. _V
13	10, 11, 12	ovmpt2a 6791	. . 3 |- ( ( C e. _V /\ H e. _V ) -> ( C |`cat H ) = ( ( C ↾s dom dom H ) sSet <. ( Hom ` ndx ) , H >. ) )
14	2, 3, 13	syl2an 494	. 2 |- ( ( C e. V /\ H e. W ) -> ( C |`cat H ) = ( ( C ↾s dom dom H ) sSet <. ( Hom ` ndx ) , H >. ) )
15	1, 14	syl5eq 2668	1 |- ( ( C e. V /\ H e. W ) -> D = ( ( C ↾s dom dom H ) sSet <. ( Hom ` ndx ) , H >. ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   dom cdm 5114   ` cfv 5888  (class class class)co 6650   ndxcnx 15854   sSet csts 15855   ↾<sub>s</sub> cress 15858   Hom chom 15952   |`_cat cresc 16468

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-resc 16471

This theorem is referenced by:  rescval2  16488