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Theorem cmp2morpcod 25965
Description: Codomain of the composite of two morphisms. (Contributed by FL, 7-Nov-2013.) (Revised by Mario Carneiro, 27-Dec-2014.)
Hypotheses
Ref Expression
cmp2morpcatt.1  |-  O  =  ( ro SetCat `  U
)
cmp2morpcatt.2  |- .Morphism  =  ( Morphism SetCat `  U )
cmp2morpcatt.3  |- .dom  =  ( dom SetCat `  U
)
cmp2morpcatt.4  |- .cod  =  ( cod SetCat `  U
)
Assertion
Ref Expression
cmp2morpcod  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  (.cod  `  ( A O B ) )  =  (.cod  `  A ) )

Proof of Theorem cmp2morpcod
StepHypRef Expression
1 simp1 955 . . 3  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  U  e.  Univ )
2 cmp2morpcatt.1 . . . 4  |-  O  =  ( ro SetCat `  U
)
3 cmp2morpcatt.2 . . . 4  |- .Morphism  =  ( Morphism SetCat `  U )
4 cmp2morpcatt.3 . . . 4  |- .dom  =  ( dom SetCat `  U
)
5 cmp2morpcatt.4 . . . 4  |- .cod  =  ( cod SetCat `  U
)
62, 3, 4, 5rocatval2 25960 . . 3  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  ( A O B )  e. .Morphism  )
73, 5codcatval2 25937 . . 3  |-  ( ( U  e.  Univ  /\  ( A O B )  e. .Morphism  )  ->  (.cod  `  ( A O B ) )  =  ( ( 2nd  o.  1st ) `  ( A O B ) ) )
81, 6, 7syl2anc 642 . 2  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  (.cod  `  ( A O B ) )  =  ( ( 2nd  o.  1st ) `  ( A O B ) ) )
92, 3, 4, 5cmp2morpcats 25961 . . . 4  |-  ( ( 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 ) ) >.
)
109fveq2d 5529 . . 3  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  ( ( 2nd  o.  1st ) `  ( A O B ) )  =  ( ( 2nd  o.  1st ) `  <. <. (.dom  `  B ) ,  (.cod  `  A )
>. ,  ( ( 2nd `  A )  o.  ( 2nd `  B
) ) >. )
)
11 fo1st 6139 . . . . . 6  |-  1st : _V -onto-> _V
12 fof 5451 . . . . . 6  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
1311, 12ax-mp 8 . . . . 5  |-  1st : _V
--> _V
14 opex 4237 . . . . 5  |-  <. <. (.dom  `  B ) ,  (.cod  `  A )
>. ,  ( ( 2nd `  A )  o.  ( 2nd `  B
) ) >.  e.  _V
15 fvco3 5596 . . . . 5  |-  ( ( 1st : _V --> _V  /\  <. <. (.dom  `  B
) ,  (.cod  `  A ) >. ,  ( ( 2nd `  A
)  o.  ( 2nd `  B ) ) >.  e.  _V )  ->  (
( 2nd  o.  1st ) `  <. <. (.dom  `  B ) ,  (.cod  `  A )
>. ,  ( ( 2nd `  A )  o.  ( 2nd `  B
) ) >. )  =  ( 2nd `  ( 1st `  <. <. (.dom  `  B ) ,  (.cod  `  A )
>. ,  ( ( 2nd `  A )  o.  ( 2nd `  B
) ) >. )
) )
1613, 14, 15mp2an 653 . . . 4  |-  ( ( 2nd  o.  1st ) `  <. <. (.dom  `  B ) ,  (.cod  `  A )
>. ,  ( ( 2nd `  A )  o.  ( 2nd `  B
) ) >. )  =  ( 2nd `  ( 1st `  <. <. (.dom  `  B ) ,  (.cod  `  A )
>. ,  ( ( 2nd `  A )  o.  ( 2nd `  B
) ) >. )
)
17 opex 4237 . . . . . 6  |-  <. (.dom  `  B ) ,  (.cod  `  A )
>.  e.  _V
18 fvex 5539 . . . . . . 7  |-  ( 2nd `  A )  e.  _V
19 fvex 5539 . . . . . . 7  |-  ( 2nd `  B )  e.  _V
2018, 19coex 5216 . . . . . 6  |-  ( ( 2nd `  A )  o.  ( 2nd `  B
) )  e.  _V
2117, 20op1st 6128 . . . . 5  |-  ( 1st `  <. <. (.dom  `  B ) ,  (.cod  `  A )
>. ,  ( ( 2nd `  A )  o.  ( 2nd `  B
) ) >. )  =  <. (.dom  `  B ) ,  (.cod  `  A )
>.
2221fveq2i 5528 . . . 4  |-  ( 2nd `  ( 1st `  <. <.
(.dom  `  B
) ,  (.cod  `  A ) >. ,  ( ( 2nd `  A
)  o.  ( 2nd `  B ) ) >.
) )  =  ( 2nd `  <. (.dom  `  B ) ,  (.cod  `  A )
>. )
23 fvex 5539 . . . . 5  |-  (.dom  `  B )  e.  _V
24 fvex 5539 . . . . 5  |-  (.cod  `  A )  e.  _V
2523, 24op2nd 6129 . . . 4  |-  ( 2nd `  <. (.dom  `  B ) ,  (.cod  `  A )
>. )  =  (.cod  `  A )
2616, 22, 253eqtri 2307 . . 3  |-  ( ( 2nd  o.  1st ) `  <. <. (.dom  `  B ) ,  (.cod  `  A )
>. ,  ( ( 2nd `  A )  o.  ( 2nd `  B
) ) >. )  =  (.cod  `  A
)
2710, 26syl6eq 2331 . 2  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  ( ( 2nd  o.  1st ) `  ( A O B ) )  =  (.cod  `  A ) )
288, 27eqtrd 2315 1  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  (.cod  `  ( A O B ) )  =  (.cod  `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643    o. ccom 4693   -->wf 5251   -onto->wfo 5253   ` 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:  cmpmorass  25966  setiscat  25979
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-map 6774  df-morcatset 25911  df-domcatset 25920  df-codcatset 25933  df-rocatset 25956
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