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Theorem resf1st 14908
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 14906 . . 3  |-  ( ph  ->  ( F  |`f  H )  =  <. ( ( 1st `  F
)  |`  dom  dom  H
) ,  ( z  e.  dom  H  |->  ( ( ( 2nd `  F
) `  z )  |`  ( H `  z
) ) ) >.
)
43fveq2d 5795 . 2  |-  ( ph  ->  ( 1st `  ( F  |`f  H ) )  =  ( 1st `  <. ( ( 1st `  F
)  |`  dom  dom  H
) ,  ( z  e.  dom  H  |->  ( ( ( 2nd `  F
) `  z )  |`  ( H `  z
) ) ) >.
) )
5 fvex 5801 . . . 4  |-  ( 1st `  F )  e.  _V
65resex 5250 . . 3  |-  ( ( 1st `  F )  |`  dom  dom  H )  e.  _V
7 dmexg 6611 . . . 4  |-  ( H  e.  W  ->  dom  H  e.  _V )
8 mptexg 6048 . . . 4  |-  ( dom 
H  e.  _V  ->  ( z  e.  dom  H  |->  ( ( ( 2nd `  F ) `  z
)  |`  ( H `  z ) ) )  e.  _V )
92, 7, 83syl 20 . . 3  |-  ( ph  ->  ( z  e.  dom  H 
|->  ( ( ( 2nd `  F ) `  z
)  |`  ( H `  z ) ) )  e.  _V )
10 op1stg 6691 . . 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 663 . 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 5610 . . . . . 6  |-  ( H  Fn  ( S  X.  S )  ->  dom  H  =  ( S  X.  S ) )
1412, 13syl 16 . . . . 5  |-  ( ph  ->  dom  H  =  ( S  X.  S ) )
1514dmeqd 5142 . . . 4  |-  ( ph  ->  dom  dom  H  =  dom  ( S  X.  S
) )
16 dmxpid 5159 . . . 4  |-  dom  ( S  X.  S )  =  S
1715, 16syl6eq 2508 . . 3  |-  ( ph  ->  dom  dom  H  =  S )
1817reseq2d 5210 . 2  |-  ( ph  ->  ( ( 1st `  F
)  |`  dom  dom  H
)  =  ( ( 1st `  F )  |`  S ) )
194, 11, 183eqtrd 2496 1  |-  ( ph  ->  ( 1st `  ( F  |`f  H ) )  =  ( ( 1st `  F
)  |`  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   _Vcvv 3070   <.cop 3983    |-> cmpt 4450    X. cxp 4938   dom cdm 4940    |` cres 4942    Fn wfn 5513   ` cfv 5518  (class class class)co 6192   1stc1st 6677   2ndc2nd 6678    |`f cresf 14871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-1st 6679  df-resf 14875
This theorem is referenced by: (None)
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