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Theorem coapm 13919
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 2296 . . . . . 6  |-  (comp `  C )  =  (comp `  C )
41, 2, 3coafval 13912 . . . . 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 5962 . . . 4  |-  Fun  .x.
6 funfn 5299 . . . 4  |-  ( Fun 
.x. 
<-> 
.x.  Fn  dom  .x.  )
75, 6mpbi 199 . . 3  |-  .x.  Fn  dom  .x.
81, 2dmcoass 13914 . . . . . . . . 9  |-  dom  .x.  C_  ( A  X.  A
)
98sseli 3189 . . . . . . . 8  |-  ( z  e.  dom  .x.  ->  z  e.  ( A  X.  A ) )
10 1st2nd2 6175 . . . . . . . 8  |-  ( z  e.  ( A  X.  A )  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
119, 10syl 15 . . . . . . 7  |-  ( z  e.  dom  .x.  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
1211fveq2d 5545 . . . . . 6  |-  ( z  e.  dom  .x.  ->  ( 
.x.  `  z )  =  (  .x.  `  <. ( 1st `  z ) ,  ( 2nd `  z
) >. ) )
13 df-ov 5877 . . . . . 6  |-  ( ( 1st `  z ) 
.x.  ( 2nd `  z
) )  =  ( 
.x.  `  <. ( 1st `  z ) ,  ( 2nd `  z )
>. )
1412, 13syl6eqr 2346 . . . . 5  |-  ( z  e.  dom  .x.  ->  ( 
.x.  `  z )  =  ( ( 1st `  z )  .x.  ( 2nd `  z ) ) )
15 eqid 2296 . . . . . . 7  |-  (Homa `  C
)  =  (Homa `  C
)
162, 15homarw 13894 . . . . . 6  |-  ( (domA `  ( 2nd `  z ) ) (Homa
`  C ) (coda `  ( 1st `  z ) ) )  C_  A
17 id 19 . . . . . . . . . . . . 13  |-  ( z  e.  dom  .x.  ->  z  e.  dom  .x.  )
1811, 17eqeltrrd 2371 . . . . . . . . . . . 12  |-  ( z  e.  dom  .x.  ->  <.
( 1st `  z
) ,  ( 2nd `  z ) >.  e.  dom  .x.  )
19 df-br 4040 . . . . . . . . . . . 12  |-  ( ( 1st `  z ) dom  .x.  ( 2nd `  z )  <->  <. ( 1st `  z ) ,  ( 2nd `  z )
>.  e.  dom  .x.  )
2018, 19sylibr 203 . . . . . . . . . . 11  |-  ( z  e.  dom  .x.  ->  ( 1st `  z ) dom  .x.  ( 2nd `  z ) )
211, 2eldmcoa 13913 . . . . . . . . . . 11  |-  ( ( 1st `  z ) dom  .x.  ( 2nd `  z )  <->  ( ( 2nd `  z )  e.  A  /\  ( 1st `  z )  e.  A  /\  (coda
`  ( 2nd `  z
) )  =  (domA `  ( 1st `  z ) ) ) )
2220, 21sylib 188 . . . . . . . . . 10  |-  ( z  e.  dom  .x.  ->  ( ( 2nd `  z
)  e.  A  /\  ( 1st `  z )  e.  A  /\  (coda `  ( 2nd `  z ) )  =  (domA `  ( 1st `  z
) ) ) )
2322simp1d 967 . . . . . . . . 9  |-  ( z  e.  dom  .x.  ->  ( 2nd `  z )  e.  A )
242, 15arwhoma 13893 . . . . . . . . 9  |-  ( ( 2nd `  z )  e.  A  ->  ( 2nd `  z )  e.  ( (domA `  ( 2nd `  z
) ) (Homa `  C
) (coda
`  ( 2nd `  z
) ) ) )
2523, 24syl 15 . . . . . . . 8  |-  ( z  e.  dom  .x.  ->  ( 2nd `  z )  e.  ( (domA `  ( 2nd `  z ) ) (Homa `  C ) (coda `  ( 2nd `  z ) ) ) )
2622simp3d 969 . . . . . . . . 9  |-  ( z  e.  dom  .x.  ->  (coda `  ( 2nd `  z ) )  =  (domA `  ( 1st `  z ) ) )
2726oveq2d 5890 . . . . . . . 8  |-  ( z  e.  dom  .x.  ->  ( (domA `  ( 2nd `  z ) ) (Homa
`  C ) (coda `  ( 2nd `  z ) ) )  =  ( (domA `  ( 2nd `  z ) ) (Homa
`  C ) (domA `  ( 1st `  z ) ) ) )
2825, 27eleqtrd 2372 . . . . . . 7  |-  ( z  e.  dom  .x.  ->  ( 2nd `  z )  e.  ( (domA `  ( 2nd `  z ) ) (Homa `  C ) (domA `  ( 1st `  z ) ) ) )
2922simp2d 968 . . . . . . . 8  |-  ( z  e.  dom  .x.  ->  ( 1st `  z )  e.  A )
302, 15arwhoma 13893 . . . . . . . 8  |-  ( ( 1st `  z )  e.  A  ->  ( 1st `  z )  e.  ( (domA `  ( 1st `  z
) ) (Homa `  C
) (coda
`  ( 1st `  z
) ) ) )
3129, 30syl 15 . . . . . . 7  |-  ( z  e.  dom  .x.  ->  ( 1st `  z )  e.  ( (domA `  ( 1st `  z ) ) (Homa `  C ) (coda `  ( 1st `  z ) ) ) )
321, 15, 28, 31coahom 13918 . . . . . 6  |-  ( z  e.  dom  .x.  ->  ( ( 1st `  z
)  .x.  ( 2nd `  z ) )  e.  ( (domA `  ( 2nd `  z
) ) (Homa `  C
) (coda
`  ( 1st `  z
) ) ) )
3316, 32sseldi 3191 . . . . 5  |-  ( z  e.  dom  .x.  ->  ( ( 1st `  z
)  .x.  ( 2nd `  z ) )  e.  A )
3414, 33eqeltrd 2370 . . . 4  |-  ( z  e.  dom  .x.  ->  ( 
.x.  `  z )  e.  A )
3534rgen 2621 . . 3  |-  A. z  e.  dom  .x.  (  .x.  `  z )  e.  A
36 ffnfv 5701 . . 3  |-  (  .x.  : dom  .x.  --> A  <->  (  .x.  Fn  dom  .x.  /\  A. z  e.  dom  .x.  (  .x.  `  z )  e.  A
) )
377, 35, 36mpbir2an 886 . 2  |-  .x.  : dom  .x.  --> A
38 fvex 5555 . . . 4  |-  (Nat `  C )  e.  _V
392, 38eqeltri 2366 . . 3  |-  A  e. 
_V
4039, 39xpex 4817 . . 3  |-  ( A  X.  A )  e. 
_V
4139, 40elpm2 6815 . 2  |-  (  .x.  e.  ( A  ^pm  ( A  X.  A ) )  <-> 
(  .x.  : dom  .x.  --> A  /\  dom  .x.  C_  ( A  X.  A ) ) )
4237, 8, 41mpbir2an 886 1  |-  .x.  e.  ( A  ^pm  ( A  X.  A ) )
Colors of variables: wff set class
Syntax hints:    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560   _Vcvv 2801    C_ wss 3165   <.cop 3656   <.cotp 3657   class class class wbr 4039    X. cxp 4703   dom cdm 4705   Fun wfun 5265    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137    ^pm cpm 6789  compcco 13236  domAcdoma 13868  codaccoda 13869  Natcarw 13870  Homachoma 13871  compaccoa 13902
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-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-ot 3663  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-1st 6138  df-2nd 6139  df-pm 6791  df-cat 13586  df-doma 13872  df-coda 13873  df-homa 13874  df-arw 13875  df-coa 13904
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