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Theorem resf2nd 13769
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 5861 . 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 13766 . . . . 5  |-  ( ph  ->  ( F  |`f  H )  =  <. ( ( 1st `  F
)  |`  dom  dom  H
) ,  ( z  e.  dom  H  |->  ( ( ( 2nd `  F
) `  z )  |`  ( H `  z
) ) ) >.
)
54fveq2d 5529 . . . 4  |-  ( ph  ->  ( 2nd `  ( F  |`f  H ) )  =  ( 2nd `  <. ( ( 1st `  F
)  |`  dom  dom  H
) ,  ( z  e.  dom  H  |->  ( ( ( 2nd `  F
) `  z )  |`  ( H `  z
) ) ) >.
) )
6 fvex 5539 . . . . . 6  |-  ( 1st `  F )  e.  _V
76resex 4995 . . . . 5  |-  ( ( 1st `  F )  |`  dom  dom  H )  e.  _V
8 dmexg 4939 . . . . . 6  |-  ( H  e.  W  ->  dom  H  e.  _V )
9 mptexg 5745 . . . . . 6  |-  ( dom 
H  e.  _V  ->  ( z  e.  dom  H  |->  ( ( ( 2nd `  F ) `  z
)  |`  ( H `  z ) ) )  e.  _V )
103, 8, 93syl 18 . . . . 5  |-  ( ph  ->  ( z  e.  dom  H 
|->  ( ( ( 2nd `  F ) `  z
)  |`  ( H `  z ) ) )  e.  _V )
11 op2ndg 6133 . . . . 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 644 . . . 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 2315 . . 3  |-  ( ph  ->  ( 2nd `  ( F  |`f  H ) )  =  ( z  e.  dom  H 
|->  ( ( ( 2nd `  F ) `  z
)  |`  ( H `  z ) ) ) )
14 simpr 447 . . . . . 6  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  z  =  <. X ,  Y >. )
1514fveq2d 5529 . . . . 5  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  ( ( 2nd `  F ) `  z
)  =  ( ( 2nd `  F ) `
 <. X ,  Y >. ) )
16 df-ov 5861 . . . . 5  |-  ( X ( 2nd `  F
) Y )  =  ( ( 2nd `  F
) `  <. X ,  Y >. )
1715, 16syl6eqr 2333 . . . 4  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  ( ( 2nd `  F ) `  z
)  =  ( X ( 2nd `  F
) Y ) )
1814fveq2d 5529 . . . . 5  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  ( H `  z )  =  ( H `  <. X ,  Y >. ) )
19 df-ov 5861 . . . . 5  |-  ( X H Y )  =  ( H `  <. X ,  Y >. )
2018, 19syl6eqr 2333 . . . 4  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  ( H `  z )  =  ( X H Y ) )
2117, 20reseq12d 4956 . . 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 4721 . . . . 5  |-  ( ( X  e.  S  /\  Y  e.  S )  -> 
<. X ,  Y >.  e.  ( S  X.  S
) )
2522, 23, 24syl2anc 642 . . . 4  |-  ( ph  -> 
<. X ,  Y >.  e.  ( S  X.  S
) )
26 resf1st.s . . . . 5  |-  ( ph  ->  H  Fn  ( S  X.  S ) )
27 fndm 5343 . . . . 5  |-  ( H  Fn  ( S  X.  S )  ->  dom  H  =  ( S  X.  S ) )
2826, 27syl 15 . . . 4  |-  ( ph  ->  dom  H  =  ( S  X.  S ) )
2925, 28eleqtrrd 2360 . . 3  |-  ( ph  -> 
<. X ,  Y >.  e. 
dom  H )
30 ovex 5883 . . . . 5  |-  ( X ( 2nd `  F
) Y )  e. 
_V
3130resex 4995 . . . 4  |-  ( ( X ( 2nd `  F
) Y )  |`  ( X H Y ) )  e.  _V
3231a1i 10 . . 3  |-  ( ph  ->  ( ( X ( 2nd `  F ) Y )  |`  ( X H Y ) )  e.  _V )
3313, 21, 29, 32fvmptd 5606 . 2  |-  ( ph  ->  ( ( 2nd `  ( F  |`f  H ) ) `  <. X ,  Y >. )  =  ( ( X ( 2nd `  F
) Y )  |`  ( X H Y ) ) )
341, 33syl5eq 2327 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 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643    e. cmpt 4077    X. cxp 4687   dom cdm 4689    |` cres 4691    Fn wfn 5250   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121    |`f cresf 13731
This theorem is referenced by:  funcres  13770
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-resf 13735
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