Theorem resfval2 16553

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 )
resfval2.g	|- ( ph -> G e. X )
resfval2.d	|- ( ph -> H Fn ( S X. S ) )

Assertion

Ref	Expression
resfval2	|- ( ph -> ( <. F , G >. |`f H ) = <. ( F |` S ) , ( x e. S , y e. S |-> ( ( x G y ) |` ( x H y ) ) ) >. )

Distinct variable groups:   x, F   x, y, G   x, H, y   ph, x   x, S, y

Allowed substitution hints:   ph( y)   F( y)   V( x, y)   W( x, y)   X( x, y)

Proof of Theorem resfval2

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opex 4932	. . . 4 |- <. F , G >. e. _V
2	1	a1i 11	. . 3 |- ( ph -> <. F , G >. e. _V )
3		resfval.d	. . 3 |- ( ph -> H e. W )
4	2, 3	resfval 16552	. 2 |- ( ph -> ( <. F , G >. |`f H ) = <. ( ( 1st ` <. F , G >. ) |` dom dom H ) , ( z e. dom H |-> ( ( ( 2nd ` <. F , G >. ) ` z ) |` ( H ` z ) ) ) >. )
5		resfval.c	. . . . 5 |- ( ph -> F e. V )
6		resfval2.g	. . . . 5 |- ( ph -> G e. X )
7		op1stg 7180	. . . . 5 |- ( ( F e. V /\ G e. X ) -> ( 1st ` <. F , G >. ) = F )
8	5, 6, 7	syl2anc 693	. . . 4 |- ( ph -> ( 1st ` <. F , G >. ) = F )
9		resfval2.d	. . . . . . 7 |- ( ph -> H Fn ( S X. S ) )
10		fndm 5990	. . . . . . 7 |- ( H Fn ( S X. S ) -> dom H = ( S X. S ) )
11	9, 10	syl 17	. . . . . 6 |- ( ph -> dom H = ( S X. S ) )
12	11	dmeqd 5326	. . . . 5 |- ( ph -> dom dom H = dom ( S X. S ) )
13		dmxpid 5345	. . . . 5 |- dom ( S X. S ) = S
14	12, 13	syl6eq 2672	. . . 4 |- ( ph -> dom dom H = S )
15	8, 14	reseq12d 5397	. . 3 |- ( ph -> ( ( 1st ` <. F , G >. ) |` dom dom H ) = ( F |` S ) )
16		op2ndg 7181	. . . . . . . 8 |- ( ( F e. V /\ G e. X ) -> ( 2nd ` <. F , G >. ) = G )
17	5, 6, 16	syl2anc 693	. . . . . . 7 |- ( ph -> ( 2nd ` <. F , G >. ) = G )
18	17	fveq1d 6193	. . . . . 6 |- ( ph -> ( ( 2nd ` <. F , G >. ) ` z ) = ( G ` z ) )
19	18	reseq1d 5395	. . . . 5 |- ( ph -> ( ( ( 2nd ` <. F , G >. ) ` z ) |` ( H ` z ) ) = ( ( G ` z ) |` ( H ` z ) ) )
20	11, 19	mpteq12dv 4733	. . . 4 |- ( ph -> ( z e. dom H |-> ( ( ( 2nd ` <. F , G >. ) ` z ) |` ( H ` z ) ) ) = ( z e. ( S X. S ) |-> ( ( G ` z ) |` ( H ` z ) ) ) )
21		fveq2 6191	. . . . . . 7 |- ( z = <. x , y >. -> ( G ` z ) = ( G ` <. x , y >. ) )
22		df-ov 6653	. . . . . . 7 |- ( x G y ) = ( G ` <. x , y >. )
23	21, 22	syl6eqr 2674	. . . . . 6 |- ( z = <. x , y >. -> ( G ` z ) = ( x G y ) )
24		fveq2 6191	. . . . . . 7 |- ( z = <. x , y >. -> ( H ` z ) = ( H ` <. x , y >. ) )
25		df-ov 6653	. . . . . . 7 |- ( x H y ) = ( H ` <. x , y >. )
26	24, 25	syl6eqr 2674	. . . . . 6 |- ( z = <. x , y >. -> ( H ` z ) = ( x H y ) )
27	23, 26	reseq12d 5397	. . . . 5 |- ( z = <. x , y >. -> ( ( G ` z ) |` ( H ` z ) ) = ( ( x G y ) |` ( x H y ) ) )
28	27	mpt2mpt 6752	. . . 4 |- ( z e. ( S X. S ) |-> ( ( G ` z ) |` ( H ` z ) ) ) = ( x e. S , y e. S |-> ( ( x G y ) |` ( x H y ) ) )
29	20, 28	syl6eq 2672	. . 3 |- ( ph -> ( z e. dom H |-> ( ( ( 2nd ` <. F , G >. ) ` z ) |` ( H ` z ) ) ) = ( x e. S , y e. S |-> ( ( x G y ) |` ( x H y ) ) ) )
30	15, 29	opeq12d 4410	. 2 |- ( ph -> <. ( ( 1st ` <. F , G >. ) |` dom dom H ) , ( z e. dom H |-> ( ( ( 2nd ` <. F , G >. ) ` z ) |` ( H ` z ) ) ) >. = <. ( F |` S ) , ( x e. S , y e. S |-> ( ( x G y ) |` ( x H y ) ) ) >. )
31	4, 30	eqtrd 2656	1 |- ( ph -> ( <. F , G >. |`f H ) = <. ( F |` S ) , ( x e. S , y e. S |-> ( ( x G y ) |` ( x H y ) ) ) >. )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   |` cres 5116   Fn wfn 5883   ` 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-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  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-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-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-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-resf 16521

This theorem is referenced by:  funcrngcsetc  41998  funcringcsetc  42035