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Theorem rocatval2 25960
Description: The composite of two morphisms in the category Set is a morphism. (Contributed by FL, 7-Nov-2013.)
Hypotheses
Ref Expression
cmp2morp.1  |-  O  =  ( ro SetCat `  U
)
rocatval2.1  |- .Morphism  =  ( Morphism SetCat `  U )
rocatval2.2  |- .dom  =  ( dom SetCat `  U
)
rocatval2.3  |- .cod  =  ( cod SetCat `  U
)
Assertion
Ref Expression
rocatval2  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  ( A O B )  e. .Morphism  )

Proof of Theorem rocatval2
StepHypRef Expression
1 rocatval2.1 . 2  |- .Morphism  =  ( Morphism SetCat `  U )
2 rocatval2.2 . . . 4  |- .dom  =  ( dom SetCat `  U
)
3 fveq1 5524 . . . . . . 7  |-  (.dom  =  ( dom SetCat `  U
)  ->  (.dom  `  A )  =  ( ( dom SetCat `  U
) `  A )
)
43eqeq1d 2291 . . . . . 6  |-  (.dom  =  ( dom SetCat `  U
)  ->  ( (.dom  `  A )  =  (.cod  `  B )  <-> 
( ( dom SetCat `  U
) `  A )  =  (.cod  `  B
) ) )
543anbi3d 1258 . . . . 5  |-  (.dom  =  ( dom SetCat `  U
)  ->  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  (.dom  `  A )  =  (.cod  `  B ) )  <->  ( U  e. 
Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U ) )  /\  ( ( dom SetCat `  U
) `  A )  =  (.cod  `  B
) ) ) )
6 rocatval2.3 . . . . . 6  |- .cod  =  ( cod SetCat `  U
)
7 fveq1 5524 . . . . . . . . 9  |-  (.cod  =  ( cod SetCat `  U
)  ->  (.cod  `  B )  =  ( ( cod SetCat `  U
) `  B )
)
87eqeq2d 2294 . . . . . . . 8  |-  (.cod  =  ( cod SetCat `  U
)  ->  ( (
( dom SetCat `  U
) `  A )  =  (.cod  `  B
)  <->  ( ( dom SetCat `
 U ) `  A )  =  ( ( cod SetCat `  U
) `  B )
) )
983anbi3d 1258 . . . . . . 7  |-  (.cod  =  ( cod SetCat `  U
)  ->  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  ( ( dom
SetCat `  U ) `  A )  =  (.cod  `  B ) )  <->  ( U  e. 
Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U ) )  /\  ( ( dom SetCat `  U
) `  A )  =  ( ( cod SetCat `
 U ) `  B ) ) ) )
10 cmp2morp.1 . . . . . . . 8  |-  O  =  ( ro SetCat `  U
)
1110rocatval 25959 . . . . . . 7  |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  ( ( dom
SetCat `  U ) `  A )  =  ( ( cod SetCat `  U
) `  B )
)  ->  ( A O B )  e.  (
Morphism
SetCat `  U ) )
129, 11syl6bi 219 . . . . . 6  |-  (.cod  =  ( cod SetCat `  U
)  ->  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  ( ( dom
SetCat `  U ) `  A )  =  (.cod  `  B ) )  ->  ( A O B )  e.  (
Morphism
SetCat `  U ) ) )
136, 12ax-mp 8 . . . . 5  |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  ( ( dom
SetCat `  U ) `  A )  =  (.cod  `  B ) )  ->  ( A O B )  e.  (
Morphism
SetCat `  U ) )
145, 13syl6bi 219 . . . 4  |-  (.dom  =  ( dom SetCat `  U
)  ->  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  ( A O B )  e.  (
Morphism
SetCat `  U ) ) )
152, 14ax-mp 8 . . 3  |-  ( ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  ( A O B )  e.  (
Morphism
SetCat `  U ) )
16 eleq2 2344 . . . . . 6  |-  (.Morphism  =  ( Morphism SetCat `  U )  ->  ( A  e. .Morphism  <->  A  e.  ( Morphism SetCat `  U )
) )
17 eleq2 2344 . . . . . 6  |-  (.Morphism  =  ( Morphism SetCat `  U )  ->  ( B  e. .Morphism  <->  B  e.  ( Morphism SetCat `  U )
) )
1816, 17anbi12d 691 . . . . 5  |-  (.Morphism  =  ( Morphism SetCat `  U )  ->  ( ( A  e. .Morphism  /\  B  e. .Morphism  )  <->  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
) ) )
19183anbi2d 1257 . . . 4  |-  (.Morphism  =  ( Morphism SetCat `  U )  ->  ( ( U  e. 
Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A
)  =  (.cod  `  B ) )  <->  ( U  e.  Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U ) )  /\  (.dom  `  A
)  =  (.cod  `  B ) ) ) )
20 eleq2 2344 . . . 4  |-  (.Morphism  =  ( Morphism SetCat `  U )  ->  ( ( A O B )  e. .Morphism  <->  ( A O B )  e.  ( Morphism SetCat `  U )
) )
2119, 20imbi12d 311 . . 3  |-  (.Morphism  =  ( Morphism SetCat `  U )  ->  ( ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  ( A O B )  e. .Morphism  ) 
<->  ( ( U  e. 
Univ  /\  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U ) )  /\  (.dom  `  A
)  =  (.cod  `  B ) )  -> 
( A O B )  e.  ( Morphism SetCat `  U ) ) ) )
2215, 21mpbiri 224 . 2  |-  (.Morphism  =  ( Morphism SetCat `  U )  ->  ( ( U  e. 
Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A
)  =  (.cod  `  B ) )  -> 
( A O B )  e. .Morphism  )
)
231, 22ax-mp 8 1  |-  ( ( U  e.  Univ  /\  ( A  e. .Morphism  /\  B  e. .Morphism  )  /\  (.dom  `  A )  =  (.cod  `  B ) )  ->  ( A O B )  e. .Morphism  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   Univcgru 8412   Morphism SetCatccmrcase 25910   dom
SetCatcdomcase 25919   cod
SetCatccodcase 25932   ro SetCatcrocase 25955
This theorem is referenced by:  cmp2morpgrp  25963  cmp2morpdom  25964  cmp2morpcod  25965  cmpmorass  25966
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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