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Theorem resf2nd 15311
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 6299 . 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 15308 . . . . 5  |-  ( ph  ->  ( F  |`f  H )  =  <. ( ( 1st `  F
)  |`  dom  dom  H
) ,  ( z  e.  dom  H  |->  ( ( ( 2nd `  F
) `  z )  |`  ( H `  z
) ) ) >.
)
54fveq2d 5876 . . . 4  |-  ( ph  ->  ( 2nd `  ( F  |`f  H ) )  =  ( 2nd `  <. ( ( 1st `  F
)  |`  dom  dom  H
) ,  ( z  e.  dom  H  |->  ( ( ( 2nd `  F
) `  z )  |`  ( H `  z
) ) ) >.
) )
6 fvex 5882 . . . . . 6  |-  ( 1st `  F )  e.  _V
76resex 5327 . . . . 5  |-  ( ( 1st `  F )  |`  dom  dom  H )  e.  _V
8 dmexg 6730 . . . . . 6  |-  ( H  e.  W  ->  dom  H  e.  _V )
9 mptexg 6143 . . . . . 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 6812 . . . . 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 2498 . . 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 5876 . . . . 5  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  ( ( 2nd `  F ) `  z
)  =  ( ( 2nd `  F ) `
 <. X ,  Y >. ) )
16 df-ov 6299 . . . . 5  |-  ( X ( 2nd `  F
) Y )  =  ( ( 2nd `  F
) `  <. X ,  Y >. )
1715, 16syl6eqr 2516 . . . 4  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  ( ( 2nd `  F ) `  z
)  =  ( X ( 2nd `  F
) Y ) )
1814fveq2d 5876 . . . . 5  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  ( H `  z )  =  ( H `  <. X ,  Y >. ) )
19 df-ov 6299 . . . . 5  |-  ( X H Y )  =  ( H `  <. X ,  Y >. )
2018, 19syl6eqr 2516 . . . 4  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  ( H `  z )  =  ( X H Y ) )
2117, 20reseq12d 5284 . . 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 5040 . . . . 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 5686 . . . . 5  |-  ( H  Fn  ( S  X.  S )  ->  dom  H  =  ( S  X.  S ) )
2826, 27syl 16 . . . 4  |-  ( ph  ->  dom  H  =  ( S  X.  S ) )
2925, 28eleqtrrd 2548 . . 3  |-  ( ph  -> 
<. X ,  Y >.  e. 
dom  H )
30 ovex 6324 . . . . 5  |-  ( X ( 2nd `  F
) Y )  e. 
_V
3130resex 5327 . . . 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 5961 . 2  |-  ( ph  ->  ( ( 2nd `  ( F  |`f  H ) ) `  <. X ,  Y >. )  =  ( ( X ( 2nd `  F
) Y )  |`  ( X H Y ) ) )
341, 33syl5eq 2510 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 1395    e. wcel 1819   _Vcvv 3109   <.cop 4038    |-> cmpt 4515    X. cxp 5006   dom cdm 5008    |` cres 5010    Fn wfn 5589   ` cfv 5594  (class class class)co 6296   1stc1st 6797   2ndc2nd 6798    |`f cresf 15273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-2nd 6800  df-resf 15277
This theorem is referenced by:  funcres  15312
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