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Theorem isgraphmrph 26026
Description: The graph of a morhism in the category Set. (Contributed by FL, 6-Nov-2013.)
Assertion
Ref Expression
isgraphmrph  |-  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )
)  ->  ( ( graph
SetCat `  U ) `  A )  =  ( 2nd `  A ) )

Proof of Theorem isgraphmrph
Dummy variables  x  a are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 5555 . . . . 5  |-  ( Morphism SetCat `  U )  e.  _V
2 mptexg 5761 . . . . 5  |-  ( (
Morphism
SetCat `  U )  e. 
_V  ->  ( a  e.  ( Morphism SetCat `  U )  |->  ( 2nd `  a
) )  e.  _V )
31, 2mp1i 11 . . . 4  |-  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )
)  ->  ( a  e.  ( Morphism SetCat `  U )  |->  ( 2nd `  a
) )  e.  _V )
4 fveq2 5541 . . . . . 6  |-  ( x  =  U  ->  ( Morphism SetCat `  x )  =  (
Morphism
SetCat `  U ) )
5 eqidd 2297 . . . . . 6  |-  ( x  =  U  ->  ( 2nd `  a )  =  ( 2nd `  a
) )
64, 5mpteq12dv 4114 . . . . 5  |-  ( x  =  U  ->  (
a  e.  ( Morphism SetCat `  x )  |->  ( 2nd `  a ) )  =  ( a  e.  (
Morphism
SetCat `  U )  |->  ( 2nd `  a ) ) )
7 df-graphcatset 26025 . . . . 5  |-  graph SetCat  =  ( x  e.  Univ  |->  ( a  e.  ( Morphism SetCat `  x
)  |->  ( 2nd `  a
) ) )
86, 7fvmptg 5616 . . . 4  |-  ( ( U  e.  Univ  /\  (
a  e.  ( Morphism SetCat `  U )  |->  ( 2nd `  a ) )  e. 
_V )  ->  ( graph
SetCat `  U )  =  ( a  e.  (
Morphism
SetCat `  U )  |->  ( 2nd `  a ) ) )
93, 8syldan 456 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )
)  ->  ( graph SetCat `  U )  =  ( a  e.  ( Morphism SetCat `  U )  |->  ( 2nd `  a ) ) )
109fveq1d 5543 . 2  |-  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )
)  ->  ( ( graph
SetCat `  U ) `  A )  =  ( ( a  e.  (
Morphism
SetCat `  U )  |->  ( 2nd `  a ) ) `  A ) )
11 simpr 447 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )
)  ->  A  e.  ( Morphism SetCat `  U )
)
12 fvex 5555 . . 3  |-  ( 2nd `  A )  e.  _V
13 fveq2 5541 . . . 4  |-  ( a  =  A  ->  ( 2nd `  a )  =  ( 2nd `  A
) )
14 eqid 2296 . . . 4  |-  ( a  e.  ( Morphism SetCat `  U
)  |->  ( 2nd `  a
) )  =  ( a  e.  ( Morphism SetCat `  U )  |->  ( 2nd `  a ) )
1513, 14fvmptg 5616 . . 3  |-  ( ( A  e.  ( Morphism SetCat `  U )  /\  ( 2nd `  A )  e. 
_V )  ->  (
( a  e.  (
Morphism
SetCat `  U )  |->  ( 2nd `  a ) ) `  A )  =  ( 2nd `  A
) )
1611, 12, 15sylancl 643 . 2  |-  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )
)  ->  ( (
a  e.  ( Morphism SetCat `  U )  |->  ( 2nd `  a ) ) `  A )  =  ( 2nd `  A ) )
1710, 16eqtrd 2328 1  |-  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )
)  ->  ( ( graph
SetCat `  U ) `  A )  =  ( 2nd `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    e. cmpt 4093   ` cfv 5271   2ndc2nd 6137   Univcgru 8428   Morphism SetCatccmrcase 26013   graph SetCatcgraphcase 26024
This theorem is referenced by:  isgraphmrph2  26027  prismorcset3  26041  grphidmor  26055  cmpidmor2  26072
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-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-pr 4230
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-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-graphcatset 26025
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