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Theorem resf2nd 14094
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.
StepHypRef Expression
1 df-ov 6086 . 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 )
42, 3resfval 14091 . . . . 5  |-  ( ph  ->  ( F  |`f  H )  =  <. ( ( 1st `  F
)  |`  dom  dom  H
) ,  ( z  e.  dom  H  |->  ( ( ( 2nd `  F
) `  z )  |`  ( H `  z
) ) ) >.
)
54fveq2d 5734 . . . 4  |-  ( ph  ->  ( 2nd `  ( F  |`f  H ) )  =  ( 2nd `  <. ( ( 1st `  F
)  |`  dom  dom  H
) ,  ( z  e.  dom  H  |->  ( ( ( 2nd `  F
) `  z )  |`  ( H `  z
) ) ) >.
) )
6 fvex 5744 . . . . . 6  |-  ( 1st `  F )  e.  _V
76resex 5188 . . . . 5  |-  ( ( 1st `  F )  |`  dom  dom  H )  e.  _V
8 dmexg 5132 . . . . . 6  |-  ( H  e.  W  ->  dom  H  e.  _V )
9 mptexg 5967 . . . . . 6  |-  ( dom 
H  e.  _V  ->  ( z  e.  dom  H  |->  ( ( ( 2nd `  F ) `  z
)  |`  ( H `  z ) ) )  e.  _V )
103, 8, 93syl 19 . . . . 5  |-  ( ph  ->  ( z  e.  dom  H 
|->  ( ( ( 2nd `  F ) `  z
)  |`  ( H `  z ) ) )  e.  _V )
11 op2ndg 6362 . . . . 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
) ) ) )
127, 10, 11sylancr 646 . . . 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
) ) ) )
135, 12eqtrd 2470 . . 3  |-  ( ph  ->  ( 2nd `  ( F  |`f  H ) )  =  ( z  e.  dom  H 
|->  ( ( ( 2nd `  F ) `  z
)  |`  ( H `  z ) ) ) )
14 simpr 449 . . . . . 6  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  z  =  <. X ,  Y >. )
1514fveq2d 5734 . . . . 5  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  ( ( 2nd `  F ) `  z
)  =  ( ( 2nd `  F ) `
 <. X ,  Y >. ) )
16 df-ov 6086 . . . . 5  |-  ( X ( 2nd `  F
) Y )  =  ( ( 2nd `  F
) `  <. X ,  Y >. )
1715, 16syl6eqr 2488 . . . 4  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  ( ( 2nd `  F ) `  z
)  =  ( X ( 2nd `  F
) Y ) )
1814fveq2d 5734 . . . . 5  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  ( H `  z )  =  ( H `  <. X ,  Y >. ) )
19 df-ov 6086 . . . . 5  |-  ( X H Y )  =  ( H `  <. X ,  Y >. )
2018, 19syl6eqr 2488 . . . 4  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  ( H `  z )  =  ( X H Y ) )
2117, 20reseq12d 5149 . . 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 4912 . . . . 5  |-  ( ( X  e.  S  /\  Y  e.  S )  -> 
<. X ,  Y >.  e.  ( S  X.  S
) )
2522, 23, 24syl2anc 644 . . . 4  |-  ( ph  -> 
<. X ,  Y >.  e.  ( S  X.  S
) )
26 resf1st.s . . . . 5  |-  ( ph  ->  H  Fn  ( S  X.  S ) )
27 fndm 5546 . . . . 5  |-  ( H  Fn  ( S  X.  S )  ->  dom  H  =  ( S  X.  S ) )
2826, 27syl 16 . . . 4  |-  ( ph  ->  dom  H  =  ( S  X.  S ) )
2925, 28eleqtrrd 2515 . . 3  |-  ( ph  -> 
<. X ,  Y >.  e. 
dom  H )
30 ovex 6108 . . . . 5  |-  ( X ( 2nd `  F
) Y )  e. 
_V
3130resex 5188 . . . 4  |-  ( ( X ( 2nd `  F
) Y )  |`  ( X H Y ) )  e.  _V
3231a1i 11 . . 3  |-  ( ph  ->  ( ( X ( 2nd `  F ) Y )  |`  ( X H Y ) )  e.  _V )
3313, 21, 29, 32fvmptd 5812 . 2  |-  ( ph  ->  ( ( 2nd `  ( F  |`f  H ) ) `  <. X ,  Y >. )  =  ( ( X ( 2nd `  F
) Y )  |`  ( X H Y ) ) )
341, 33syl5eq 2482 1  |-  ( ph  ->  ( X ( 2nd `  ( F  |`f  H ) ) Y )  =  ( ( X ( 2nd `  F
) Y )  |`  ( X H Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   <.cop 3819    e. cmpt 4268    X. cxp 4878   dom cdm 4880    |` cres 4882    Fn wfn 5451   ` cfv 5456  (class class class)co 6083   1stc1st 6349   2ndc2nd 6350    |`f cresf 14056
This theorem is referenced by:  funcres  14095
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-2nd 6352  df-resf 14060
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