MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  coapm Unicode version

Theorem coapm 14155
Description: Composition of arrows is a partial binary operation on arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
coapm.o  |-  .x.  =  (compa `  C )
coapm.a  |-  A  =  (Nat `  C )
Assertion
Ref Expression
coapm  |-  .x.  e.  ( A  ^pm  ( A  X.  A ) )

Proof of Theorem coapm
Dummy variables  f 
g  h  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 coapm.o . . . . . 6  |-  .x.  =  (compa `  C )
2 coapm.a . . . . . 6  |-  A  =  (Nat `  C )
3 eqid 2389 . . . . . 6  |-  (comp `  C )  =  (comp `  C )
41, 2, 3coafval 14148 . . . . 5  |-  .x.  =  ( g  e.  A ,  f  e.  { h  e.  A  |  (coda `  h
)  =  (domA `  g ) }  |->  <. (domA `  f ) ,  (coda `  g ) ,  ( ( 2nd `  g
) ( <. (domA `  f ) ,  (domA `  g ) >. (comp `  C ) (coda `  g
) ) ( 2nd `  f ) ) >.
)
54mpt2fun 6113 . . . 4  |-  Fun  .x.
6 funfn 5424 . . . 4  |-  ( Fun 
.x. 
<-> 
.x.  Fn  dom  .x.  )
75, 6mpbi 200 . . 3  |-  .x.  Fn  dom  .x.
81, 2dmcoass 14150 . . . . . . . . 9  |-  dom  .x.  C_  ( A  X.  A
)
98sseli 3289 . . . . . . . 8  |-  ( z  e.  dom  .x.  ->  z  e.  ( A  X.  A ) )
10 1st2nd2 6327 . . . . . . . 8  |-  ( z  e.  ( A  X.  A )  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
119, 10syl 16 . . . . . . 7  |-  ( z  e.  dom  .x.  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
1211fveq2d 5674 . . . . . 6  |-  ( z  e.  dom  .x.  ->  ( 
.x.  `  z )  =  (  .x.  `  <. ( 1st `  z ) ,  ( 2nd `  z
) >. ) )
13 df-ov 6025 . . . . . 6  |-  ( ( 1st `  z ) 
.x.  ( 2nd `  z
) )  =  ( 
.x.  `  <. ( 1st `  z ) ,  ( 2nd `  z )
>. )
1412, 13syl6eqr 2439 . . . . 5  |-  ( z  e.  dom  .x.  ->  ( 
.x.  `  z )  =  ( ( 1st `  z )  .x.  ( 2nd `  z ) ) )
15 eqid 2389 . . . . . . 7  |-  (Homa `  C
)  =  (Homa `  C
)
162, 15homarw 14130 . . . . . 6  |-  ( (domA `  ( 2nd `  z ) ) (Homa
`  C ) (coda `  ( 1st `  z ) ) )  C_  A
17 id 20 . . . . . . . . . . . . 13  |-  ( z  e.  dom  .x.  ->  z  e.  dom  .x.  )
1811, 17eqeltrrd 2464 . . . . . . . . . . . 12  |-  ( z  e.  dom  .x.  ->  <.
( 1st `  z
) ,  ( 2nd `  z ) >.  e.  dom  .x.  )
19 df-br 4156 . . . . . . . . . . . 12  |-  ( ( 1st `  z ) dom  .x.  ( 2nd `  z )  <->  <. ( 1st `  z ) ,  ( 2nd `  z )
>.  e.  dom  .x.  )
2018, 19sylibr 204 . . . . . . . . . . 11  |-  ( z  e.  dom  .x.  ->  ( 1st `  z ) dom  .x.  ( 2nd `  z ) )
211, 2eldmcoa 14149 . . . . . . . . . . 11  |-  ( ( 1st `  z ) dom  .x.  ( 2nd `  z )  <->  ( ( 2nd `  z )  e.  A  /\  ( 1st `  z )  e.  A  /\  (coda
`  ( 2nd `  z
) )  =  (domA `  ( 1st `  z ) ) ) )
2220, 21sylib 189 . . . . . . . . . 10  |-  ( z  e.  dom  .x.  ->  ( ( 2nd `  z
)  e.  A  /\  ( 1st `  z )  e.  A  /\  (coda `  ( 2nd `  z ) )  =  (domA `  ( 1st `  z
) ) ) )
2322simp1d 969 . . . . . . . . 9  |-  ( z  e.  dom  .x.  ->  ( 2nd `  z )  e.  A )
242, 15arwhoma 14129 . . . . . . . . 9  |-  ( ( 2nd `  z )  e.  A  ->  ( 2nd `  z )  e.  ( (domA `  ( 2nd `  z
) ) (Homa `  C
) (coda
`  ( 2nd `  z
) ) ) )
2523, 24syl 16 . . . . . . . 8  |-  ( z  e.  dom  .x.  ->  ( 2nd `  z )  e.  ( (domA `  ( 2nd `  z ) ) (Homa `  C ) (coda `  ( 2nd `  z ) ) ) )
2622simp3d 971 . . . . . . . . 9  |-  ( z  e.  dom  .x.  ->  (coda `  ( 2nd `  z ) )  =  (domA `  ( 1st `  z ) ) )
2726oveq2d 6038 . . . . . . . 8  |-  ( z  e.  dom  .x.  ->  ( (domA `  ( 2nd `  z ) ) (Homa
`  C ) (coda `  ( 2nd `  z ) ) )  =  ( (domA `  ( 2nd `  z ) ) (Homa
`  C ) (domA `  ( 1st `  z ) ) ) )
2825, 27eleqtrd 2465 . . . . . . 7  |-  ( z  e.  dom  .x.  ->  ( 2nd `  z )  e.  ( (domA `  ( 2nd `  z ) ) (Homa `  C ) (domA `  ( 1st `  z ) ) ) )
2922simp2d 970 . . . . . . . 8  |-  ( z  e.  dom  .x.  ->  ( 1st `  z )  e.  A )
302, 15arwhoma 14129 . . . . . . . 8  |-  ( ( 1st `  z )  e.  A  ->  ( 1st `  z )  e.  ( (domA `  ( 1st `  z
) ) (Homa `  C
) (coda
`  ( 1st `  z
) ) ) )
3129, 30syl 16 . . . . . . 7  |-  ( z  e.  dom  .x.  ->  ( 1st `  z )  e.  ( (domA `  ( 1st `  z ) ) (Homa `  C ) (coda `  ( 1st `  z ) ) ) )
321, 15, 28, 31coahom 14154 . . . . . 6  |-  ( z  e.  dom  .x.  ->  ( ( 1st `  z
)  .x.  ( 2nd `  z ) )  e.  ( (domA `  ( 2nd `  z
) ) (Homa `  C
) (coda
`  ( 1st `  z
) ) ) )
3316, 32sseldi 3291 . . . . 5  |-  ( z  e.  dom  .x.  ->  ( ( 1st `  z
)  .x.  ( 2nd `  z ) )  e.  A )
3414, 33eqeltrd 2463 . . . 4  |-  ( z  e.  dom  .x.  ->  ( 
.x.  `  z )  e.  A )
3534rgen 2716 . . 3  |-  A. z  e.  dom  .x.  (  .x.  `  z )  e.  A
36 ffnfv 5835 . . 3  |-  (  .x.  : dom  .x.  --> A  <->  (  .x.  Fn  dom  .x.  /\  A. z  e.  dom  .x.  (  .x.  `  z )  e.  A
) )
377, 35, 36mpbir2an 887 . 2  |-  .x.  : dom  .x.  --> A
38 fvex 5684 . . . 4  |-  (Nat `  C )  e.  _V
392, 38eqeltri 2459 . . 3  |-  A  e. 
_V
4039, 39xpex 4932 . . 3  |-  ( A  X.  A )  e. 
_V
4139, 40elpm2 6983 . 2  |-  (  .x.  e.  ( A  ^pm  ( A  X.  A ) )  <-> 
(  .x.  : dom  .x.  --> A  /\  dom  .x.  C_  ( A  X.  A ) ) )
4237, 8, 41mpbir2an 887 1  |-  .x.  e.  ( A  ^pm  ( A  X.  A ) )
Colors of variables: wff set class
Syntax hints:    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2651   {crab 2655   _Vcvv 2901    C_ wss 3265   <.cop 3762   <.cotp 3763   class class class wbr 4155    X. cxp 4818   dom cdm 4820   Fun wfun 5390    Fn wfn 5391   -->wf 5392   ` cfv 5396  (class class class)co 6022   1stc1st 6288   2ndc2nd 6289    ^pm cpm 6957  compcco 13470  domAcdoma 14104  codaccoda 14105  Natcarw 14106  Homachoma 14107  compaccoa 14138
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-ot 3769  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-pm 6959  df-cat 13822  df-doma 14108  df-coda 14109  df-homa 14110  df-arw 14111  df-coa 14140
  Copyright terms: Public domain W3C validator