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Theorem resf2nd 15122
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 6287 . 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 15119 . . . . 5  |-  ( ph  ->  ( F  |`f  H )  =  <. ( ( 1st `  F
)  |`  dom  dom  H
) ,  ( z  e.  dom  H  |->  ( ( ( 2nd `  F
) `  z )  |`  ( H `  z
) ) ) >.
)
54fveq2d 5870 . . . 4  |-  ( ph  ->  ( 2nd `  ( F  |`f  H ) )  =  ( 2nd `  <. ( ( 1st `  F
)  |`  dom  dom  H
) ,  ( z  e.  dom  H  |->  ( ( ( 2nd `  F
) `  z )  |`  ( H `  z
) ) ) >.
) )
6 fvex 5876 . . . . . 6  |-  ( 1st `  F )  e.  _V
76resex 5317 . . . . 5  |-  ( ( 1st `  F )  |`  dom  dom  H )  e.  _V
8 dmexg 6715 . . . . . 6  |-  ( H  e.  W  ->  dom  H  e.  _V )
9 mptexg 6130 . . . . . 6  |-  ( dom 
H  e.  _V  ->  ( z  e.  dom  H  |->  ( ( ( 2nd `  F ) `  z
)  |`  ( H `  z ) ) )  e.  _V )
103, 8, 93syl 20 . . . . 5  |-  ( ph  ->  ( z  e.  dom  H 
|->  ( ( ( 2nd `  F ) `  z
)  |`  ( H `  z ) ) )  e.  _V )
11 op2ndg 6797 . . . . 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 663 . . . 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 2508 . . 3  |-  ( ph  ->  ( 2nd `  ( F  |`f  H ) )  =  ( z  e.  dom  H 
|->  ( ( ( 2nd `  F ) `  z
)  |`  ( H `  z ) ) ) )
14 simpr 461 . . . . . 6  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  z  =  <. X ,  Y >. )
1514fveq2d 5870 . . . . 5  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  ( ( 2nd `  F ) `  z
)  =  ( ( 2nd `  F ) `
 <. X ,  Y >. ) )
16 df-ov 6287 . . . . 5  |-  ( X ( 2nd `  F
) Y )  =  ( ( 2nd `  F
) `  <. X ,  Y >. )
1715, 16syl6eqr 2526 . . . 4  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  ( ( 2nd `  F ) `  z
)  =  ( X ( 2nd `  F
) Y ) )
1814fveq2d 5870 . . . . 5  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  ( H `  z )  =  ( H `  <. X ,  Y >. ) )
19 df-ov 6287 . . . . 5  |-  ( X H Y )  =  ( H `  <. X ,  Y >. )
2018, 19syl6eqr 2526 . . . 4  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  ( H `  z )  =  ( X H Y ) )
2117, 20reseq12d 5274 . . 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 5031 . . . . 5  |-  ( ( X  e.  S  /\  Y  e.  S )  -> 
<. X ,  Y >.  e.  ( S  X.  S
) )
2522, 23, 24syl2anc 661 . . . 4  |-  ( ph  -> 
<. X ,  Y >.  e.  ( S  X.  S
) )
26 resf1st.s . . . . 5  |-  ( ph  ->  H  Fn  ( S  X.  S ) )
27 fndm 5680 . . . . 5  |-  ( H  Fn  ( S  X.  S )  ->  dom  H  =  ( S  X.  S ) )
2826, 27syl 16 . . . 4  |-  ( ph  ->  dom  H  =  ( S  X.  S ) )
2925, 28eleqtrrd 2558 . . 3  |-  ( ph  -> 
<. X ,  Y >.  e. 
dom  H )
30 ovex 6309 . . . . 5  |-  ( X ( 2nd `  F
) Y )  e. 
_V
3130resex 5317 . . . 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 5955 . 2  |-  ( ph  ->  ( ( 2nd `  ( F  |`f  H ) ) `  <. X ,  Y >. )  =  ( ( X ( 2nd `  F
) Y )  |`  ( X H Y ) ) )
341, 33syl5eq 2520 1  |-  ( ph  ->  ( X ( 2nd `  ( F  |`f  H ) ) Y )  =  ( ( X ( 2nd `  F
) Y )  |`  ( X H Y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113   <.cop 4033    |-> cmpt 4505    X. cxp 4997   dom cdm 4999    |` cres 5001    Fn wfn 5583   ` cfv 5588  (class class class)co 6284   1stc1st 6782   2ndc2nd 6783    |`f cresf 15084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-2nd 6785  df-resf 15088
This theorem is referenced by:  funcres  15123
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