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Theorem resf1st 15811
Description: Value of the functor restriction operator on objects. (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 ) )
Assertion
Ref Expression
resf1st  |-  ( ph  ->  ( 1st `  ( F  |`f  H ) )  =  ( ( 1st `  F
)  |`  S ) )

Proof of Theorem resf1st
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 resf1st.f . . . 4  |-  ( ph  ->  F  e.  V )
2 resf1st.h . . . 4  |-  ( ph  ->  H  e.  W )
31, 2resfval 15809 . . 3  |-  ( ph  ->  ( F  |`f  H )  =  <. ( ( 1st `  F
)  |`  dom  dom  H
) ,  ( z  e.  dom  H  |->  ( ( ( 2nd `  F
) `  z )  |`  ( H `  z
) ) ) >.
)
43fveq2d 5874 . 2  |-  ( ph  ->  ( 1st `  ( F  |`f  H ) )  =  ( 1st `  <. ( ( 1st `  F
)  |`  dom  dom  H
) ,  ( z  e.  dom  H  |->  ( ( ( 2nd `  F
) `  z )  |`  ( H `  z
) ) ) >.
) )
5 fvex 5880 . . . 4  |-  ( 1st `  F )  e.  _V
65resex 5151 . . 3  |-  ( ( 1st `  F )  |`  dom  dom  H )  e.  _V
7 dmexg 6729 . . . 4  |-  ( H  e.  W  ->  dom  H  e.  _V )
8 mptexg 6140 . . . 4  |-  ( dom 
H  e.  _V  ->  ( z  e.  dom  H  |->  ( ( ( 2nd `  F ) `  z
)  |`  ( H `  z ) ) )  e.  _V )
92, 7, 83syl 18 . . 3  |-  ( ph  ->  ( z  e.  dom  H 
|->  ( ( ( 2nd `  F ) `  z
)  |`  ( H `  z ) ) )  e.  _V )
10 op1stg 6810 . . 3  |-  ( ( ( ( 1st `  F
)  |`  dom  dom  H
)  e.  _V  /\  ( z  e.  dom  H 
|->  ( ( ( 2nd `  F ) `  z
)  |`  ( H `  z ) ) )  e.  _V )  -> 
( 1st `  <. ( ( 1st `  F
)  |`  dom  dom  H
) ,  ( z  e.  dom  H  |->  ( ( ( 2nd `  F
) `  z )  |`  ( H `  z
) ) ) >.
)  =  ( ( 1st `  F )  |`  dom  dom  H )
)
116, 9, 10sylancr 670 . 2  |-  ( ph  ->  ( 1st `  <. ( ( 1st `  F
)  |`  dom  dom  H
) ,  ( z  e.  dom  H  |->  ( ( ( 2nd `  F
) `  z )  |`  ( H `  z
) ) ) >.
)  =  ( ( 1st `  F )  |`  dom  dom  H )
)
12 resf1st.s . . . . . 6  |-  ( ph  ->  H  Fn  ( S  X.  S ) )
13 fndm 5680 . . . . . 6  |-  ( H  Fn  ( S  X.  S )  ->  dom  H  =  ( S  X.  S ) )
1412, 13syl 17 . . . . 5  |-  ( ph  ->  dom  H  =  ( S  X.  S ) )
1514dmeqd 5040 . . . 4  |-  ( ph  ->  dom  dom  H  =  dom  ( S  X.  S
) )
16 dmxpid 5057 . . . 4  |-  dom  ( S  X.  S )  =  S
1715, 16syl6eq 2503 . . 3  |-  ( ph  ->  dom  dom  H  =  S )
1817reseq2d 5108 . 2  |-  ( ph  ->  ( ( 1st `  F
)  |`  dom  dom  H
)  =  ( ( 1st `  F )  |`  S ) )
194, 11, 183eqtrd 2491 1  |-  ( ph  ->  ( 1st `  ( F  |`f  H ) )  =  ( ( 1st `  F
)  |`  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1446    e. wcel 1889   _Vcvv 3047   <.cop 3976    |-> cmpt 4464    X. cxp 4835   dom cdm 4837    |` cres 4839    Fn wfn 5580   ` cfv 5585  (class class class)co 6295   1stc1st 6796   2ndc2nd 6797    |`f cresf 15774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6798  df-resf 15778
This theorem is referenced by: (None)
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