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Theorem cmppar3 25974
Description: Morphisms composition is defined every time the codomain of the second operand matches the domain of the first one. (Contributed by FL, 8-Nov-2013.)
Assertion
Ref Expression
cmppar3  |-  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U ) )  -> 
( <. A ,  B >.  e.  dom  ( ro SetCat `
 U )  <->  ( ( dom
SetCat `  U ) `  A )  =  ( ( cod SetCat `  U
) `  B )
) )

Proof of Theorem cmppar3
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cmppar2 25972 . . . 4  |-  ( U  e.  Univ  ->  dom  ( ro SetCat `  U )  =  { <. a ,  b
>.  |  ( a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )  /\  (
( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
) ) } )
213ad2ant1 976 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U ) )  ->  dom  ( ro SetCat `  U
)  =  { <. a ,  b >.  |  ( a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )  /\  ( ( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
) ) } )
32eleq2d 2350 . 2  |-  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U ) )  -> 
( <. A ,  B >.  e.  dom  ( ro SetCat `
 U )  <->  <. A ,  B >.  e.  { <. a ,  b >.  |  ( a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )  /\  ( ( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
) ) } ) )
4 elex 2796 . . . 4  |-  ( A  e.  ( Morphism SetCat `  U
)  ->  A  e.  _V )
543ad2ant2 977 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U ) )  ->  A  e.  _V )
6 elex 2796 . . . 4  |-  ( B  e.  ( Morphism SetCat `  U
)  ->  B  e.  _V )
763ad2ant3 978 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U ) )  ->  B  e.  _V )
8 eleq1 2343 . . . . 5  |-  ( a  =  A  ->  (
a  e.  ( Morphism SetCat `  U )  <->  A  e.  ( Morphism SetCat `  U )
) )
9 fveq2 5525 . . . . . 6  |-  ( a  =  A  ->  (
( 1st  o.  1st ) `  a )  =  ( ( 1st 
o.  1st ) `  A
) )
109eqeq1d 2291 . . . . 5  |-  ( a  =  A  ->  (
( ( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
)  <->  ( ( 1st 
o.  1st ) `  A
)  =  ( ( 2nd  o.  1st ) `  b ) ) )
118, 103anbi13d 1254 . . . 4  |-  ( a  =  A  ->  (
( a  e.  (
Morphism
SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )  /\  (
( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
) )  <->  ( A  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )  /\  (
( 1st  o.  1st ) `  A )  =  ( ( 2nd 
o.  1st ) `  b
) ) ) )
12 eleq1 2343 . . . . 5  |-  ( b  =  B  ->  (
b  e.  ( Morphism SetCat `  U )  <->  B  e.  ( Morphism SetCat `  U )
) )
13 fveq2 5525 . . . . . 6  |-  ( b  =  B  ->  (
( 2nd  o.  1st ) `  b )  =  ( ( 2nd 
o.  1st ) `  B
) )
1413eqeq2d 2294 . . . . 5  |-  ( b  =  B  ->  (
( ( 1st  o.  1st ) `  A )  =  ( ( 2nd 
o.  1st ) `  b
)  <->  ( ( 1st 
o.  1st ) `  A
)  =  ( ( 2nd  o.  1st ) `  B ) ) )
1512, 143anbi23d 1255 . . . 4  |-  ( b  =  B  ->  (
( A  e.  (
Morphism
SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )  /\  (
( 1st  o.  1st ) `  A )  =  ( ( 2nd 
o.  1st ) `  b
) )  <->  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )  /\  (
( 1st  o.  1st ) `  A )  =  ( ( 2nd 
o.  1st ) `  B
) ) ) )
1611, 15opelopabg 4283 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( <. A ,  B >.  e.  { <. a ,  b >.  |  ( a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )  /\  ( ( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
) ) }  <->  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )  /\  (
( 1st  o.  1st ) `  A )  =  ( ( 2nd 
o.  1st ) `  B
) ) ) )
175, 7, 16syl2anc 642 . 2  |-  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U ) )  -> 
( <. A ,  B >.  e.  { <. a ,  b >.  |  ( a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )  /\  ( ( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
) ) }  <->  ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )  /\  (
( 1st  o.  1st ) `  A )  =  ( ( 2nd 
o.  1st ) `  B
) ) ) )
18 ibar 490 . . . . 5  |-  ( ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  ->  ( (
( 1st  o.  1st ) `  A )  =  ( ( 2nd 
o.  1st ) `  B
)  <->  ( ( A  e.  ( Morphism SetCat `  U
)  /\  B  e.  ( Morphism SetCat `  U )
)  /\  ( ( 1st  o.  1st ) `  A )  =  ( ( 2nd  o.  1st ) `  B )
) ) )
19183adant1 973 . . . 4  |-  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U ) )  -> 
( ( ( 1st 
o.  1st ) `  A
)  =  ( ( 2nd  o.  1st ) `  B )  <->  ( ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  ( ( 1st  o.  1st ) `  A )  =  ( ( 2nd  o.  1st ) `  B )
) ) )
20 df-3an 936 . . . 4  |-  ( ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )  /\  ( ( 1st  o.  1st ) `  A )  =  ( ( 2nd 
o.  1st ) `  B
) )  <->  ( ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )
)  /\  ( ( 1st  o.  1st ) `  A )  =  ( ( 2nd  o.  1st ) `  B )
) )
2119, 20syl6rbbr 255 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U ) )  -> 
( ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )  /\  (
( 1st  o.  1st ) `  A )  =  ( ( 2nd 
o.  1st ) `  B
) )  <->  ( ( 1st  o.  1st ) `  A )  =  ( ( 2nd  o.  1st ) `  B )
) )
22 domcatval 25930 . . . . . 6  |-  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )
)  ->  ( ( dom
SetCat `  U ) `  A )  =  ( ( 1st  o.  1st ) `  A )
)
23223adant3 975 . . . . 5  |-  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U ) )  -> 
( ( dom SetCat `  U
) `  A )  =  ( ( 1st 
o.  1st ) `  A
) )
2423eqcomd 2288 . . . 4  |-  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U ) )  -> 
( ( 1st  o.  1st ) `  A )  =  ( ( dom SetCat `
 U ) `  A ) )
25 codcatval 25936 . . . . . 6  |-  ( ( U  e.  Univ  /\  B  e.  ( Morphism SetCat `  U )
)  ->  ( ( cod
SetCat `  U ) `  B )  =  ( ( 2nd  o.  1st ) `  B )
)
26253adant2 974 . . . . 5  |-  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U ) )  -> 
( ( cod SetCat `  U
) `  B )  =  ( ( 2nd 
o.  1st ) `  B
) )
2726eqcomd 2288 . . . 4  |-  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U ) )  -> 
( ( 2nd  o.  1st ) `  B )  =  ( ( cod SetCat `
 U ) `  B ) )
2824, 27eqeq12d 2297 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U ) )  -> 
( ( ( 1st 
o.  1st ) `  A
)  =  ( ( 2nd  o.  1st ) `  B )  <->  ( ( dom
SetCat `  U ) `  A )  =  ( ( cod SetCat `  U
) `  B )
) )
2921, 28bitrd 244 . 2  |-  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U ) )  -> 
( ( A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U )  /\  (
( 1st  o.  1st ) `  A )  =  ( ( 2nd 
o.  1st ) `  B
) )  <->  ( ( dom
SetCat `  U ) `  A )  =  ( ( cod SetCat `  U
) `  B )
) )
303, 17, 293bitrd 270 1  |-  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )  /\  B  e.  ( Morphism SetCat `  U ) )  -> 
( <. A ,  B >.  e.  dom  ( ro SetCat `
 U )  <->  ( ( dom
SetCat `  U ) `  A )  =  ( ( cod SetCat `  U
) `  B )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643   {copab 4076   dom cdm 4689    o. ccom 4693   ` cfv 5255   1stc1st 6120   2ndc2nd 6121   Univcgru 8412   Morphism SetCatccmrcase 25910   dom
SetCatcdomcase 25919   cod
SetCatccodcase 25932   ro SetCatcrocase 25955
This theorem is referenced by:  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-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-domcatset 25920  df-codcatset 25933  df-rocatset 25956
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