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Theorem morcatset1 25915
Description: The morphisms of the category Set. (Contributed by FL, 6-Nov-2013.)
Assertion
Ref Expression
morcatset1  |-  ( U  e.  Univ  ->  ( Morphism SetCat `  U )  =  { <. <. a ,  b
>. ,  c >.  |  ( a  e.  U  /\  b  e.  U  /\  c  e.  (
b  ^m  a )
) } )
Distinct variable group:    U, a, b, c

Proof of Theorem morcatset1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 prismorcsetlem 25912 . 2  |-  ( U  e.  Univ  ->  { <. <.
a ,  b >. ,  c >.  |  ( a  e.  U  /\  b  e.  U  /\  c  e.  ( b  ^m  a ) ) }  e.  _V )
2 eleq2 2344 . . . . 5  |-  ( x  =  U  ->  (
a  e.  x  <->  a  e.  U ) )
3 eleq2 2344 . . . . 5  |-  ( x  =  U  ->  (
b  e.  x  <->  b  e.  U ) )
42, 33anbi12d 1253 . . . 4  |-  ( x  =  U  ->  (
( a  e.  x  /\  b  e.  x  /\  c  e.  (
b  ^m  a )
)  <->  ( a  e.  U  /\  b  e.  U  /\  c  e.  ( b  ^m  a
) ) ) )
54oprabbidv 5902 . . 3  |-  ( x  =  U  ->  { <. <.
a ,  b >. ,  c >.  |  ( a  e.  x  /\  b  e.  x  /\  c  e.  ( b  ^m  a ) ) }  =  { <. <. a ,  b >. ,  c
>.  |  ( a  e.  U  /\  b  e.  U  /\  c  e.  ( b  ^m  a
) ) } )
6 df-morcatset 25911 . . 3  |-  Morphism SetCat  =  ( x  e.  Univ  |->  { <. <.
a ,  b >. ,  c >.  |  ( a  e.  x  /\  b  e.  x  /\  c  e.  ( b  ^m  a ) ) } )
75, 6fvmptg 5600 . 2  |-  ( ( U  e.  Univ  /\  { <. <. a ,  b
>. ,  c >.  |  ( a  e.  U  /\  b  e.  U  /\  c  e.  (
b  ^m  a )
) }  e.  _V )  ->  ( Morphism SetCat `  U
)  =  { <. <.
a ,  b >. ,  c >.  |  ( a  e.  U  /\  b  e.  U  /\  c  e.  ( b  ^m  a ) ) } )
81, 7mpdan 649 1  |-  ( U  e.  Univ  ->  ( Morphism SetCat `  U )  =  { <. <. a ,  b
>. ,  c >.  |  ( a  e.  U  /\  b  e.  U  /\  c  e.  (
b  ^m  a )
) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788   ` cfv 5255  (class class class)co 5858   {coprab 5859    ^m cmap 6772   Univcgru 8412   Morphism SetCatccmrcase 25910
This theorem is referenced by:  prismorcset2  25918  morexcmp  25967
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-morcatset 25911
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