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Theorem resf2nd 13785
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 5877 . 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 13782 . . . . 5  |-  ( ph  ->  ( F  |`f  H )  =  <. ( ( 1st `  F
)  |`  dom  dom  H
) ,  ( z  e.  dom  H  |->  ( ( ( 2nd `  F
) `  z )  |`  ( H `  z
) ) ) >.
)
54fveq2d 5545 . . . 4  |-  ( ph  ->  ( 2nd `  ( F  |`f  H ) )  =  ( 2nd `  <. ( ( 1st `  F
)  |`  dom  dom  H
) ,  ( z  e.  dom  H  |->  ( ( ( 2nd `  F
) `  z )  |`  ( H `  z
) ) ) >.
) )
6 fvex 5555 . . . . . 6  |-  ( 1st `  F )  e.  _V
76resex 5011 . . . . 5  |-  ( ( 1st `  F )  |`  dom  dom  H )  e.  _V
8 dmexg 4955 . . . . . 6  |-  ( H  e.  W  ->  dom  H  e.  _V )
9 mptexg 5761 . . . . . 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 6149 . . . . 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 2328 . . 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 5545 . . . . 5  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  ( ( 2nd `  F ) `  z
)  =  ( ( 2nd `  F ) `
 <. X ,  Y >. ) )
16 df-ov 5877 . . . . 5  |-  ( X ( 2nd `  F
) Y )  =  ( ( 2nd `  F
) `  <. X ,  Y >. )
1715, 16syl6eqr 2346 . . . 4  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  ( ( 2nd `  F ) `  z
)  =  ( X ( 2nd `  F
) Y ) )
1814fveq2d 5545 . . . . 5  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  ( H `  z )  =  ( H `  <. X ,  Y >. ) )
19 df-ov 5877 . . . . 5  |-  ( X H Y )  =  ( H `  <. X ,  Y >. )
2018, 19syl6eqr 2346 . . . 4  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  ( H `  z )  =  ( X H Y ) )
2117, 20reseq12d 4972 . . 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 4737 . . . . 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 5359 . . . . 5  |-  ( H  Fn  ( S  X.  S )  ->  dom  H  =  ( S  X.  S ) )
2826, 27syl 15 . . . 4  |-  ( ph  ->  dom  H  =  ( S  X.  S ) )
2925, 28eleqtrrd 2373 . . 3  |-  ( ph  -> 
<. X ,  Y >.  e. 
dom  H )
30 ovex 5899 . . . . 5  |-  ( X ( 2nd `  F
) Y )  e. 
_V
3130resex 5011 . . . 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 5622 . 2  |-  ( ph  ->  ( ( 2nd `  ( F  |`f  H ) ) `  <. X ,  Y >. )  =  ( ( X ( 2nd `  F
) Y )  |`  ( X H Y ) ) )
341, 33syl5eq 2340 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 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656    e. cmpt 4093    X. cxp 4703   dom cdm 4705    |` cres 4707    Fn wfn 5266   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137    |`f cresf 13747
This theorem is referenced by:  funcres  13786
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-resf 13751
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