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Theorem resf1st 14091
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 14089 . . 3  |-  ( ph  ->  ( F  |`f  H )  =  <. ( ( 1st `  F
)  |`  dom  dom  H
) ,  ( z  e.  dom  H  |->  ( ( ( 2nd `  F
) `  z )  |`  ( H `  z
) ) ) >.
)
43fveq2d 5732 . 2  |-  ( ph  ->  ( 1st `  ( F  |`f  H ) )  =  ( 1st `  <. ( ( 1st `  F
)  |`  dom  dom  H
) ,  ( z  e.  dom  H  |->  ( ( ( 2nd `  F
) `  z )  |`  ( H `  z
) ) ) >.
) )
5 fvex 5742 . . . 4  |-  ( 1st `  F )  e.  _V
65resex 5186 . . 3  |-  ( ( 1st `  F )  |`  dom  dom  H )  e.  _V
7 dmexg 5130 . . . 4  |-  ( H  e.  W  ->  dom  H  e.  _V )
8 mptexg 5965 . . . 4  |-  ( dom 
H  e.  _V  ->  ( z  e.  dom  H  |->  ( ( ( 2nd `  F ) `  z
)  |`  ( H `  z ) ) )  e.  _V )
92, 7, 83syl 19 . . 3  |-  ( ph  ->  ( z  e.  dom  H 
|->  ( ( ( 2nd `  F ) `  z
)  |`  ( H `  z ) ) )  e.  _V )
10 op1stg 6359 . . 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 645 . 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 5544 . . . . . 6  |-  ( H  Fn  ( S  X.  S )  ->  dom  H  =  ( S  X.  S ) )
1412, 13syl 16 . . . . 5  |-  ( ph  ->  dom  H  =  ( S  X.  S ) )
1514dmeqd 5072 . . . 4  |-  ( ph  ->  dom  dom  H  =  dom  ( S  X.  S
) )
16 dmxpid 5089 . . . 4  |-  dom  ( S  X.  S )  =  S
1715, 16syl6eq 2484 . . 3  |-  ( ph  ->  dom  dom  H  =  S )
1817reseq2d 5146 . 2  |-  ( ph  ->  ( ( 1st `  F
)  |`  dom  dom  H
)  =  ( ( 1st `  F )  |`  S ) )
194, 11, 183eqtrd 2472 1  |-  ( ph  ->  ( 1st `  ( F  |`f  H ) )  =  ( ( 1st `  F
)  |`  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2956   <.cop 3817    e. cmpt 4266    X. cxp 4876   dom cdm 4878    |` cres 4880    Fn wfn 5449   ` cfv 5454  (class class class)co 6081   1stc1st 6347   2ndc2nd 6348    |`f cresf 14054
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-resf 14058
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