Theorem resfval 16552

Description: Value of the functor restriction operator. (Contributed by Mario Carneiro, 6-Jan-2017.)

Hypotheses

Ref	Expression
resfval.c	|- ( ph -> F e. V )
resfval.d	|- ( ph -> H e. W )

Assertion

Ref	Expression
resfval	|- ( ph -> ( F |`f H ) = <. ( ( 1st ` F ) |` dom dom H ) , ( x e. dom H |-> ( ( ( 2nd ` F ) ` x ) |` ( H ` x ) ) ) >. )

Distinct variable groups:   x, F   x, H   ph, x

Allowed substitution hints:   V( x)   W( x)

Proof of Theorem resfval

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

Step	Hyp	Ref	Expression
1		df-resf 16521	. . 3 |- |`f = ( f e. _V , h e. _V |-> <. ( ( 1st ` f ) |` dom dom h ) , ( x e. dom h |-> ( ( ( 2nd ` f ) ` x ) |` ( h ` x ) ) ) >. )
2	1	a1i 11	. 2 |- ( ph -> |`f = ( f e. _V , h e. _V |-> <. ( ( 1st ` f ) |` dom dom h ) , ( x e. dom h |-> ( ( ( 2nd ` f ) ` x ) |` ( h ` x ) ) ) >. ) )
3		simprl 794	. . . . 5 |- ( ( ph /\ ( f = F /\ h = H ) ) -> f = F )
4	3	fveq2d 6195	. . . 4 |- ( ( ph /\ ( f = F /\ h = H ) ) -> ( 1st ` f ) = ( 1st ` F ) )
5		simprr 796	. . . . . 6 |- ( ( ph /\ ( f = F /\ h = H ) ) -> h = H )
6	5	dmeqd 5326	. . . . 5 |- ( ( ph /\ ( f = F /\ h = H ) ) -> dom h = dom H )
7	6	dmeqd 5326	. . . 4 |- ( ( ph /\ ( f = F /\ h = H ) ) -> dom dom h = dom dom H )
8	4, 7	reseq12d 5397	. . 3 |- ( ( ph /\ ( f = F /\ h = H ) ) -> ( ( 1st ` f ) |` dom dom h ) = ( ( 1st ` F ) |` dom dom H ) )
9	3	fveq2d 6195	. . . . . 6 |- ( ( ph /\ ( f = F /\ h = H ) ) -> ( 2nd ` f ) = ( 2nd ` F ) )
10	9	fveq1d 6193	. . . . 5 |- ( ( ph /\ ( f = F /\ h = H ) ) -> ( ( 2nd ` f ) ` x ) = ( ( 2nd ` F ) ` x ) )
11	5	fveq1d 6193	. . . . 5 |- ( ( ph /\ ( f = F /\ h = H ) ) -> ( h ` x ) = ( H ` x ) )
12	10, 11	reseq12d 5397	. . . 4 |- ( ( ph /\ ( f = F /\ h = H ) ) -> ( ( ( 2nd ` f ) ` x ) |` ( h ` x ) ) = ( ( ( 2nd ` F ) ` x ) |` ( H ` x ) ) )
13	6, 12	mpteq12dv 4733	. . 3 |- ( ( ph /\ ( f = F /\ h = H ) ) -> ( x e. dom h |-> ( ( ( 2nd ` f ) ` x ) |` ( h ` x ) ) ) = ( x e. dom H |-> ( ( ( 2nd ` F ) ` x ) |` ( H ` x ) ) ) )
14	8, 13	opeq12d 4410	. 2 |- ( ( ph /\ ( f = F /\ h = H ) ) -> <. ( ( 1st ` f ) |` dom dom h ) , ( x e. dom h |-> ( ( ( 2nd ` f ) ` x ) |` ( h ` x ) ) ) >. = <. ( ( 1st ` F ) |` dom dom H ) , ( x e. dom H |-> ( ( ( 2nd ` F ) ` x ) |` ( H ` x ) ) ) >. )
15		resfval.c	. . 3 |- ( ph -> F e. V )
16		elex 3212	. . 3 |- ( F e. V -> F e. _V )
17	15, 16	syl 17	. 2 |- ( ph -> F e. _V )
18		resfval.d	. . 3 |- ( ph -> H e. W )
19		elex 3212	. . 3 |- ( H e. W -> H e. _V )
20	18, 19	syl 17	. 2 |- ( ph -> H e. _V )
21		opex 4932	. . 3 |- <. ( ( 1st ` F ) |` dom dom H ) , ( x e. dom H |-> ( ( ( 2nd ` F ) ` x ) |` ( H ` x ) ) ) >. e. _V
22	21	a1i 11	. 2 |- ( ph -> <. ( ( 1st ` F ) |` dom dom H ) , ( x e. dom H |-> ( ( ( 2nd ` F ) ` x ) |` ( H ` x ) ) ) >. e. _V )
23	2, 14, 17, 20, 22	ovmpt2d 6788	1 |- ( ph -> ( F |`f H ) = <. ( ( 1st ` F ) |` dom dom H ) , ( x e. dom H |-> ( ( ( 2nd ` F ) ` x ) |` ( H ` x ) ) ) >. )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   |-> cmpt 4729   dom cdm 5114   |` cres 5116   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   |`_f cresf 16517

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-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-resf 16521

This theorem is referenced by:  resfval2  16553  resf1st  16554  resf2nd  16555  funcres  16556