Theorem resf2nd 16555

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

Hypotheses

Ref	Expression
resf1st.f	|- ( ph -> F e. V )
resf1st.h	|- ( ph -> H e. W )
resf1st.s	|- ( ph -> H Fn ( S X. S ) )
resf2nd.x	|- ( ph -> X e. S )
resf2nd.y	|- ( ph -> Y e. S )

Assertion

Ref	Expression
resf2nd	|- ( ph -> ( X ( 2nd ` ( F |`f H ) ) Y ) = ( ( X ( 2nd ` F ) Y ) |` ( X H Y ) ) )

Proof of Theorem resf2nd

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ov 6653	. 2 |- ( X ( 2nd ` ( F |`f H ) ) Y ) = ( ( 2nd ` ( F |`f H ) ) ` <. X , Y >. )
2		resf1st.f	. . . . . 6 |- ( ph -> F e. V )
3		resf1st.h	. . . . . 6 |- ( ph -> H e. W )
4	2, 3	resfval 16552	. . . . 5 |- ( ph -> ( F |`f H ) = <. ( ( 1st ` F ) |` dom dom H ) , ( z e. dom H |-> ( ( ( 2nd ` F ) ` z ) |` ( H ` z ) ) ) >. )
5	4	fveq2d 6195	. . . 4 |- ( ph -> ( 2nd ` ( F |`f H ) ) = ( 2nd ` <. ( ( 1st ` F ) |` dom dom H ) , ( z e. dom H |-> ( ( ( 2nd ` F ) ` z ) |` ( H ` z ) ) ) >. ) )
6		fvex 6201	. . . . . 6 |- ( 1st ` F ) e. _V
7	6	resex 5443	. . . . 5 |- ( ( 1st ` F ) |` dom dom H ) e. _V
8		dmexg 7097	. . . . . 6 |- ( H e. W -> dom H e. _V )
9		mptexg 6484	. . . . . 6 |- ( dom H e. _V -> ( z e. dom H |-> ( ( ( 2nd ` F ) ` z ) |` ( H ` z ) ) ) e. _V )
10	3, 8, 9	3syl 18	. . . . 5 |- ( ph -> ( z e. dom H |-> ( ( ( 2nd ` F ) ` z ) |` ( H ` z ) ) ) e. _V )
11		op2ndg 7181	. . . . 5 |- ( ( ( ( 1st ` F ) |` dom dom H ) e. _V /\ ( z e. dom H |-> ( ( ( 2nd ` F ) ` z ) |` ( H ` z ) ) ) e. _V ) -> ( 2nd ` <. ( ( 1st ` F ) |` dom dom H ) , ( z e. dom H |-> ( ( ( 2nd ` F ) ` z ) |` ( H ` z ) ) ) >. ) = ( z e. dom H |-> ( ( ( 2nd ` F ) ` z ) |` ( H ` z ) ) ) )
12	7, 10, 11	sylancr 695	. . . 4 |- ( ph -> ( 2nd ` <. ( ( 1st ` F ) |` dom dom H ) , ( z e. dom H |-> ( ( ( 2nd ` F ) ` z ) |` ( H ` z ) ) ) >. ) = ( z e. dom H |-> ( ( ( 2nd ` F ) ` z ) |` ( H ` z ) ) ) )
13	5, 12	eqtrd 2656	. . 3 |- ( ph -> ( 2nd ` ( F |`f H ) ) = ( z e. dom H |-> ( ( ( 2nd ` F ) ` z ) |` ( H ` z ) ) ) )
14		simpr 477	. . . . . 6 |- ( ( ph /\ z = <. X , Y >. ) -> z = <. X , Y >. )
15	14	fveq2d 6195	. . . . 5 |- ( ( ph /\ z = <. X , Y >. ) -> ( ( 2nd ` F ) ` z ) = ( ( 2nd ` F ) ` <. X , Y >. ) )
16		df-ov 6653	. . . . 5 |- ( X ( 2nd ` F ) Y ) = ( ( 2nd ` F ) ` <. X , Y >. )
17	15, 16	syl6eqr 2674	. . . 4 |- ( ( ph /\ z = <. X , Y >. ) -> ( ( 2nd ` F ) ` z ) = ( X ( 2nd ` F ) Y ) )
18	14	fveq2d 6195	. . . . 5 |- ( ( ph /\ z = <. X , Y >. ) -> ( H ` z ) = ( H ` <. X , Y >. ) )
19		df-ov 6653	. . . . 5 |- ( X H Y ) = ( H ` <. X , Y >. )
20	18, 19	syl6eqr 2674	. . . 4 |- ( ( ph /\ z = <. X , Y >. ) -> ( H ` z ) = ( X H Y ) )
21	17, 20	reseq12d 5397	. . 3 |- ( ( ph /\ z = <. X , Y >. ) -> ( ( ( 2nd ` F ) ` z ) |` ( H ` z ) ) = ( ( X ( 2nd ` F ) Y ) |` ( X H Y ) ) )
22		resf2nd.x	. . . . 5 |- ( ph -> X e. S )
23		resf2nd.y	. . . . 5 |- ( ph -> Y e. S )
24		opelxpi 5148	. . . . 5 |- ( ( X e. S /\ Y e. S ) -> <. X , Y >. e. ( S X. S ) )
25	22, 23, 24	syl2anc 693	. . . 4 |- ( ph -> <. X , Y >. e. ( S X. S ) )
26		resf1st.s	. . . . 5 |- ( ph -> H Fn ( S X. S ) )
27		fndm 5990	. . . . 5 |- ( H Fn ( S X. S ) -> dom H = ( S X. S ) )
28	26, 27	syl 17	. . . 4 |- ( ph -> dom H = ( S X. S ) )
29	25, 28	eleqtrrd 2704	. . 3 |- ( ph -> <. X , Y >. e. dom H )
30		ovex 6678	. . . . 5 |- ( X ( 2nd ` F ) Y ) e. _V
31	30	resex 5443	. . . 4 |- ( ( X ( 2nd ` F ) Y ) |` ( X H Y ) ) e. _V
32	31	a1i 11	. . 3 |- ( ph -> ( ( X ( 2nd ` F ) Y ) |` ( X H Y ) ) e. _V )
33	13, 21, 29, 32	fvmptd 6288	. 2 |- ( ph -> ( ( 2nd ` ( F |`f H ) ) ` <. X , Y >. ) = ( ( X ( 2nd ` F ) Y ) |` ( X H Y ) ) )
34	1, 33	syl5eq 2668	1 |- ( ph -> ( X ( 2nd ` ( F |`f H ) ) Y ) = ( ( X ( 2nd ` F ) Y ) |` ( X H Y ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = 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   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-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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-2nd 7169  df-resf 16521

This theorem is referenced by:  funcres  16556