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Theorem cmpmorfun 26074
Description: Composition of morphisms is a function. (Contributed by FL, 8-Nov-2013.) (Revised by Mario Carneiro, 20-Dec-2013.)
Assertion
Ref Expression
cmpmorfun  |-  ( U  e.  Univ  ->  Fun  ( ro SetCat `  U )
)

Proof of Theorem cmpmorfun
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 moeq 2954 . . . 4  |-  E* c 
c  =  <. <. (
( 1st  o.  1st ) `  b ) ,  ( ( 2nd 
o.  1st ) `  a
) >. ,  ( ( 2nd `  a )  o.  ( 2nd `  b
) ) >.
21moani 2208 . . 3  |-  E* c
( ( a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )  /\  (
( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
) )  /\  c  =  <. <. ( ( 1st 
o.  1st ) `  b
) ,  ( ( 2nd  o.  1st ) `  a ) >. ,  ( ( 2nd `  a
)  o.  ( 2nd `  b ) ) >.
)
32funoprab 5960 . 2  |-  Fun  { <. <. a ,  b
>. ,  c >.  |  ( ( a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )  /\  (
( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
) )  /\  c  =  <. <. ( ( 1st 
o.  1st ) `  b
) ,  ( ( 2nd  o.  1st ) `  a ) >. ,  ( ( 2nd `  a
)  o.  ( 2nd `  b ) ) >.
) }
4 isrocatset 26060 . . 3  |-  ( U  e.  Univ  ->  ( ro SetCat `
 U )  =  { <. <. a ,  b
>. ,  c >.  |  ( ( a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )  /\  (
( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
) )  /\  c  =  <. <. ( ( 1st 
o.  1st ) `  b
) ,  ( ( 2nd  o.  1st ) `  a ) >. ,  ( ( 2nd `  a
)  o.  ( 2nd `  b ) ) >.
) } )
54funeqd 5292 . 2  |-  ( U  e.  Univ  ->  ( Fun  ( ro SetCat `  U
)  <->  Fun  { <. <. a ,  b >. ,  c
>.  |  ( (
a  e.  ( Morphism SetCat `  U )  /\  b  e.  ( Morphism SetCat `  U )  /\  ( ( 1st  o.  1st ) `  a )  =  ( ( 2nd 
o.  1st ) `  b
) )  /\  c  =  <. <. ( ( 1st 
o.  1st ) `  b
) ,  ( ( 2nd  o.  1st ) `  a ) >. ,  ( ( 2nd `  a
)  o.  ( 2nd `  b ) ) >.
) } ) )
63, 5mpbiri 224 1  |-  ( U  e.  Univ  ->  Fun  ( ro SetCat `  U )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   <.cop 3656    o. ccom 4709   Fun wfun 5265   ` cfv 5271   {coprab 5875   1stc1st 6136   2ndc2nd 6137   Univcgru 8428   Morphism SetCatccmrcase 26013   ro SetCatcrocase 26058
This theorem is referenced by:  cmpmor  26078  setiscat  26082
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-rocatset 26059
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