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Theorem resf1st 13817
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 13815 . . 3  |-  ( ph  ->  ( F  |`f  H )  =  <. ( ( 1st `  F
)  |`  dom  dom  H
) ,  ( z  e.  dom  H  |->  ( ( ( 2nd `  F
) `  z )  |`  ( H `  z
) ) ) >.
)
43fveq2d 5567 . 2  |-  ( ph  ->  ( 1st `  ( F  |`f  H ) )  =  ( 1st `  <. ( ( 1st `  F
)  |`  dom  dom  H
) ,  ( z  e.  dom  H  |->  ( ( ( 2nd `  F
) `  z )  |`  ( H `  z
) ) ) >.
) )
5 fvex 5577 . . . 4  |-  ( 1st `  F )  e.  _V
65resex 5032 . . 3  |-  ( ( 1st `  F )  |`  dom  dom  H )  e.  _V
7 dmexg 4976 . . . 4  |-  ( H  e.  W  ->  dom  H  e.  _V )
8 mptexg 5786 . . . 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 6174 . . 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 644 . 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 5380 . . . . . 6  |-  ( H  Fn  ( S  X.  S )  ->  dom  H  =  ( S  X.  S ) )
1412, 13syl 15 . . . . 5  |-  ( ph  ->  dom  H  =  ( S  X.  S ) )
1514dmeqd 4918 . . . 4  |-  ( ph  ->  dom  dom  H  =  dom  ( S  X.  S
) )
16 dmxpid 4935 . . . 4  |-  dom  ( S  X.  S )  =  S
1715, 16syl6eq 2364 . . 3  |-  ( ph  ->  dom  dom  H  =  S )
1817reseq2d 4992 . 2  |-  ( ph  ->  ( ( 1st `  F
)  |`  dom  dom  H
)  =  ( ( 1st `  F )  |`  S ) )
194, 11, 183eqtrd 2352 1  |-  ( ph  ->  ( 1st `  ( F  |`f  H ) )  =  ( ( 1st `  F
)  |`  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1633    e. wcel 1701   _Vcvv 2822   <.cop 3677    e. cmpt 4114    X. cxp 4724   dom cdm 4726    |` cres 4728    Fn wfn 5287   ` cfv 5292  (class class class)co 5900   1stc1st 6162   2ndc2nd 6163    |`f cresf 13780
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-resf 13784
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