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Theorem coapm 15256
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 2467 . . . . . 6  |-  (comp `  C )  =  (comp `  C )
41, 2, 3coafval 15249 . . . . 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 6388 . . . 4  |-  Fun  .x.
6 funfn 5617 . . . 4  |-  ( Fun 
.x. 
<-> 
.x.  Fn  dom  .x.  )
75, 6mpbi 208 . . 3  |-  .x.  Fn  dom  .x.
81, 2dmcoass 15251 . . . . . . . . 9  |-  dom  .x.  C_  ( A  X.  A
)
98sseli 3500 . . . . . . . 8  |-  ( z  e.  dom  .x.  ->  z  e.  ( A  X.  A ) )
10 1st2nd2 6821 . . . . . . . 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 5870 . . . . . 6  |-  ( z  e.  dom  .x.  ->  ( 
.x.  `  z )  =  (  .x.  `  <. ( 1st `  z ) ,  ( 2nd `  z
) >. ) )
13 df-ov 6287 . . . . . 6  |-  ( ( 1st `  z ) 
.x.  ( 2nd `  z
) )  =  ( 
.x.  `  <. ( 1st `  z ) ,  ( 2nd `  z )
>. )
1412, 13syl6eqr 2526 . . . . 5  |-  ( z  e.  dom  .x.  ->  ( 
.x.  `  z )  =  ( ( 1st `  z )  .x.  ( 2nd `  z ) ) )
15 eqid 2467 . . . . . . 7  |-  (Homa `  C
)  =  (Homa `  C
)
162, 15homarw 15231 . . . . . 6  |-  ( (domA `  ( 2nd `  z ) ) (Homa
`  C ) (coda `  ( 1st `  z ) ) )  C_  A
17 id 22 . . . . . . . . . . . . 13  |-  ( z  e.  dom  .x.  ->  z  e.  dom  .x.  )
1811, 17eqeltrrd 2556 . . . . . . . . . . . 12  |-  ( z  e.  dom  .x.  ->  <.
( 1st `  z
) ,  ( 2nd `  z ) >.  e.  dom  .x.  )
19 df-br 4448 . . . . . . . . . . . 12  |-  ( ( 1st `  z ) dom  .x.  ( 2nd `  z )  <->  <. ( 1st `  z ) ,  ( 2nd `  z )
>.  e.  dom  .x.  )
2018, 19sylibr 212 . . . . . . . . . . 11  |-  ( z  e.  dom  .x.  ->  ( 1st `  z ) dom  .x.  ( 2nd `  z ) )
211, 2eldmcoa 15250 . . . . . . . . . . 11  |-  ( ( 1st `  z ) dom  .x.  ( 2nd `  z )  <->  ( ( 2nd `  z )  e.  A  /\  ( 1st `  z )  e.  A  /\  (coda
`  ( 2nd `  z
) )  =  (domA `  ( 1st `  z ) ) ) )
2220, 21sylib 196 . . . . . . . . . 10  |-  ( z  e.  dom  .x.  ->  ( ( 2nd `  z
)  e.  A  /\  ( 1st `  z )  e.  A  /\  (coda `  ( 2nd `  z ) )  =  (domA `  ( 1st `  z
) ) ) )
2322simp1d 1008 . . . . . . . . 9  |-  ( z  e.  dom  .x.  ->  ( 2nd `  z )  e.  A )
242, 15arwhoma 15230 . . . . . . . . 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 1010 . . . . . . . . 9  |-  ( z  e.  dom  .x.  ->  (coda `  ( 2nd `  z ) )  =  (domA `  ( 1st `  z ) ) )
2726oveq2d 6300 . . . . . . . 8  |-  ( z  e.  dom  .x.  ->  ( (domA `  ( 2nd `  z ) ) (Homa
`  C ) (coda `  ( 2nd `  z ) ) )  =  ( (domA `  ( 2nd `  z ) ) (Homa
`  C ) (domA `  ( 1st `  z ) ) ) )
2825, 27eleqtrd 2557 . . . . . . 7  |-  ( z  e.  dom  .x.  ->  ( 2nd `  z )  e.  ( (domA `  ( 2nd `  z ) ) (Homa `  C ) (domA `  ( 1st `  z ) ) ) )
2922simp2d 1009 . . . . . . . 8  |-  ( z  e.  dom  .x.  ->  ( 1st `  z )  e.  A )
302, 15arwhoma 15230 . . . . . . . 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 15255 . . . . . 6  |-  ( z  e.  dom  .x.  ->  ( ( 1st `  z
)  .x.  ( 2nd `  z ) )  e.  ( (domA `  ( 2nd `  z
) ) (Homa `  C
) (coda
`  ( 1st `  z
) ) ) )
3316, 32sseldi 3502 . . . . 5  |-  ( z  e.  dom  .x.  ->  ( ( 1st `  z
)  .x.  ( 2nd `  z ) )  e.  A )
3414, 33eqeltrd 2555 . . . 4  |-  ( z  e.  dom  .x.  ->  ( 
.x.  `  z )  e.  A )
3534rgen 2824 . . 3  |-  A. z  e.  dom  .x.  (  .x.  `  z )  e.  A
36 ffnfv 6047 . . 3  |-  (  .x.  : dom  .x.  --> A  <->  (  .x.  Fn  dom  .x.  /\  A. z  e.  dom  .x.  (  .x.  `  z )  e.  A
) )
377, 35, 36mpbir2an 918 . 2  |-  .x.  : dom  .x.  --> A
38 fvex 5876 . . . 4  |-  (Nat `  C )  e.  _V
392, 38eqeltri 2551 . . 3  |-  A  e. 
_V
4039, 39xpex 6588 . . 3  |-  ( A  X.  A )  e. 
_V
4139, 40elpm2 7450 . 2  |-  (  .x.  e.  ( A  ^pm  ( A  X.  A ) )  <-> 
(  .x.  : dom  .x.  --> A  /\  dom  .x.  C_  ( A  X.  A ) ) )
4237, 8, 41mpbir2an 918 1  |-  .x.  e.  ( A  ^pm  ( A  X.  A ) )
Colors of variables: wff setvar class
Syntax hints:    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   {crab 2818   _Vcvv 3113    C_ wss 3476   <.cop 4033   <.cotp 4035   class class class wbr 4447    X. cxp 4997   dom cdm 4999   Fun wfun 5582    Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6284   1stc1st 6782   2ndc2nd 6783    ^pm cpm 7421  compcco 14567  domAcdoma 15205  codaccoda 15206  Natcarw 15207  Homachoma 15208  compaccoa 15239
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-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-ot 4036  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-1st 6784  df-2nd 6785  df-pm 7423  df-cat 14923  df-doma 15209  df-coda 15210  df-homa 15211  df-arw 15212  df-coa 15241
This theorem is referenced by: (None)
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