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Theorem cmp2morpcats 25961
Description: Composite of two morphisms. (Contributed by FL, 7-Nov-2013.)
Hypotheses
Ref Expression
cmp2morpcats.1  |-  O  =  ( ro SetCat `  U
)
cmp2morpcats.2  |- .Morphism  =  ( Morphism SetCat `  U )
cmp2morpcats.3  |- .dom  =  ( dom SetCat `  U
)
cmp2morpcats.4  |- .cod  =  ( cod SetCat `  U
)
Assertion
Ref Expression
cmp2morpcats  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  ( A O B )  =  <. <.
(.dom  `  B
) ,  (.cod  `  A ) >. ,  ( ( 2nd `  A
)  o.  ( 2nd `  B ) ) >.
)

Proof of Theorem cmp2morpcats
StepHypRef Expression
1 simp1 955 . . 3  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  U  e.  Univ )
2 cmp2morpcats.2 . . . . . . 7  |- .Morphism  =  ( Morphism SetCat `  U )
32eleq2i 2347 . . . . . 6  |-  ( A  e. .Morphism  <->  A  e.  ( Morphism SetCat `  U )
)
43biimpi 186 . . . . 5  |-  ( A  e. .Morphism  ->  A  e.  ( Morphism SetCat `  U )
)
54adantr 451 . . . 4  |-  ( ( A  e. .Morphism  /\  B  e. .Morphism  )  ->  A  e.  ( Morphism SetCat `  U ) )
653ad2ant2 977 . . 3  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  A  e.  ( Morphism SetCat `  U )
)
72eleq2i 2347 . . . . . 6  |-  ( B  e. .Morphism  <->  B  e.  ( Morphism SetCat `  U )
)
87biimpi 186 . . . . 5  |-  ( B  e. .Morphism  ->  B  e.  ( Morphism SetCat `  U )
)
98adantl 452 . . . 4  |-  ( ( A  e. .Morphism  /\  B  e. .Morphism  )  ->  B  e.  ( Morphism SetCat `  U ) )
1093ad2ant2 977 . . 3  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  B  e.  ( Morphism SetCat `  U )
)
11 cmp2morpcats.3 . . . . . . 7  |- .dom  =  ( dom SetCat `  U
)
1211fveq1i 5526 . . . . . 6  |-  (.dom  `  A )  =  ( ( dom SetCat `  U
) `  A )
13 cmp2morpcats.4 . . . . . . 7  |- .cod  =  ( cod SetCat `  U
)
1413fveq1i 5526 . . . . . 6  |-  (.cod  `  B )  =  ( ( cod SetCat `  U
) `  B )
1512, 14eqeq12i 2296 . . . . 5  |-  ( (.dom  `  A )  =  (.cod  `  B )  <->  ( ( dom
SetCat `  U ) `  A )  =  ( ( cod SetCat `  U
) `  B )
)
1615biimpi 186 . . . 4  |-  ( (.dom  `  A )  =  (.cod  `  B )  ->  (
( dom SetCat `  U
) `  A )  =  ( ( cod SetCat `
 U ) `  B ) )
17163ad2ant3 978 . . 3  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  ( ( dom
SetCat `  U ) `  A )  =  ( ( cod SetCat `  U
) `  B )
)
18 cmp2morpcats.1 . . . 4  |-  O  =  ( ro SetCat `  U
)
1918cmp2morp 25958 . . 3  |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  ( ( dom
SetCat `  U ) `  A )  =  ( ( cod SetCat `  U
) `  B )
)  ->  ( A O B )  =  <. <.
( ( 1st  o.  1st ) `  B ) ,  ( ( 2nd 
o.  1st ) `  A
) >. ,  ( ( 2nd `  A )  o.  ( 2nd `  B
) ) >. )
201, 6, 10, 17, 19syl121anc 1187 . 2  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  ( A O B )  =  <. <.
( ( 1st  o.  1st ) `  B ) ,  ( ( 2nd 
o.  1st ) `  A
) >. ,  ( ( 2nd `  A )  o.  ( 2nd `  B
) ) >. )
218ad2antll 709 . . . . . . . . 9  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  ) )  ->  B  e.  ( Morphism SetCat `  U ) )
22 domcatval 25930 . . . . . . . . 9  |-  ( ( U  e.  Univ  /\  B  e.  ( Morphism SetCat `  U )
)  ->  ( ( dom
SetCat `  U ) `  B )  =  ( ( 1st  o.  1st ) `  B )
)
2321, 22syldan 456 . . . . . . . 8  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  ) )  -> 
( ( dom SetCat `  U
) `  B )  =  ( ( 1st 
o.  1st ) `  B
) )
2423eqcomd 2288 . . . . . . 7  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  ) )  -> 
( ( 1st  o.  1st ) `  B )  =  ( ( dom SetCat `
 U ) `  B ) )
25 fveq1 5524 . . . . . . . 8  |-  (.dom  =  ( dom SetCat `  U
)  ->  (.dom  `  B )  =  ( ( dom SetCat `  U
) `  B )
)
2625eqeq2d 2294 . . . . . . 7  |-  (.dom  =  ( dom SetCat `  U
)  ->  ( (
( 1st  o.  1st ) `  B )  =  (.dom  `  B
)  <->  ( ( 1st 
o.  1st ) `  B
)  =  ( ( dom SetCat `  U ) `  B ) ) )
2724, 26syl5ibr 212 . . . . . 6  |-  (.dom  =  ( dom SetCat `  U
)  ->  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  ) )  -> 
( ( 1st  o.  1st ) `  B )  =  (.dom  `  B ) ) )
2811, 27ax-mp 8 . . . . 5  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  ) )  -> 
( ( 1st  o.  1st ) `  B )  =  (.dom  `  B ) )
29283adant3 975 . . . 4  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  ( ( 1st  o.  1st ) `  B )  =  (.dom  `  B ) )
305anim2i 552 . . . . . . . . 9  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  ) )  -> 
( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U ) ) )
31303adant3 975 . . . . . . . 8  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )
) )
32 codcatval 25936 . . . . . . . 8  |-  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )
)  ->  ( ( cod
SetCat `  U ) `  A )  =  ( ( 2nd  o.  1st ) `  A )
)
3331, 32syl 15 . . . . . . 7  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  ( ( cod
SetCat `  U ) `  A )  =  ( ( 2nd  o.  1st ) `  A )
)
34 fveq1 5524 . . . . . . . 8  |-  (.cod  =  ( cod SetCat `  U
)  ->  (.cod  `  A )  =  ( ( cod SetCat `  U
) `  A )
)
3534eqeq1d 2291 . . . . . . 7  |-  (.cod  =  ( cod SetCat `  U
)  ->  ( (.cod  `  A )  =  ( ( 2nd  o.  1st ) `  A )  <->  ( ( cod SetCat `  U
) `  A )  =  ( ( 2nd 
o.  1st ) `  A
) ) )
3633, 35syl5ibr 212 . . . . . 6  |-  (.cod  =  ( cod SetCat `  U
)  ->  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  (.cod  `  A )  =  ( ( 2nd  o.  1st ) `  A )
) )
3713, 36ax-mp 8 . . . . 5  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  (.cod  `  A )  =  ( ( 2nd  o.  1st ) `  A )
)
3837eqcomd 2288 . . . 4  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  ( ( 2nd  o.  1st ) `  A )  =  (.cod  `  A ) )
3929, 38opeq12d 3804 . . 3  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  <. ( ( 1st  o.  1st ) `  B ) ,  ( ( 2nd  o.  1st ) `  A ) >.  =  <. (.dom  `  B ) ,  (.cod  `  A )
>. )
4039opeq1d 3802 . 2  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  <. <. (
( 1st  o.  1st ) `  B ) ,  ( ( 2nd 
o.  1st ) `  A
) >. ,  ( ( 2nd `  A )  o.  ( 2nd `  B
) ) >.  =  <. <.
(.dom  `  B
) ,  (.cod  `  A ) >. ,  ( ( 2nd `  A
)  o.  ( 2nd `  B ) ) >.
)
4120, 40eqtrd 2315 1  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  ( A O B )  =  <. <.
(.dom  `  B
) ,  (.cod  `  A ) >. ,  ( ( 2nd `  A
)  o.  ( 2nd `  B ) ) >.
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   <.cop 3643    o. ccom 4693   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   Univcgru 8412   Morphism SetCatccmrcase 25910   dom
SetCatcdomcase 25919   cod
SetCatccodcase 25932   ro SetCatcrocase 25955
This theorem is referenced by:  cmp2morpcatt  25962  cmp2morpdom  25964  cmp2morpcod  25965  cmpidmor2  25969
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-domcatset 25920  df-codcatset 25933  df-rocatset 25956
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