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Theorem isgraphmrph2 25924
Description: The graph of a morphism in the category Set. (Contributed by FL, 6-Nov-2013.)
Hypotheses
Ref Expression
isgraphmrph2.1  |- .graph  =  ( graph SetCat `  U )
isgraphmrph2.2  |- .Morphism  =  ( Morphism SetCat `  U )
Assertion
Ref Expression
isgraphmrph2  |-  ( ( U  e.  Univ  /\  A  e. .Morphism  )  ->  (.graph  `  A )  =  ( 2nd `  A ) )

Proof of Theorem isgraphmrph2
StepHypRef Expression
1 isgraphmrph2.1 . 2  |- .graph  =  ( graph SetCat `  U )
2 isgraphmrph2.2 . . . 4  |- .Morphism  =  ( Morphism SetCat `  U )
3 eleq2 2344 . . . . . 6  |-  (.Morphism  =  ( Morphism SetCat `  U )  ->  ( A  e. .Morphism  <->  A  e.  ( Morphism SetCat `  U )
) )
43anbi2d 684 . . . . 5  |-  (.Morphism  =  ( Morphism SetCat `  U )  ->  ( ( U  e. 
Univ  /\  A  e. .Morphism  ) 
<->  ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U ) ) ) )
5 isgraphmrph 25923 . . . . 5  |-  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )
)  ->  ( ( graph
SetCat `  U ) `  A )  =  ( 2nd `  A ) )
64, 5syl6bi 219 . . . 4  |-  (.Morphism  =  ( Morphism SetCat `  U )  ->  ( ( U  e. 
Univ  /\  A  e. .Morphism  )  ->  ( ( graph SetCat `  U ) `  A
)  =  ( 2nd `  A ) ) )
72, 6ax-mp 8 . . 3  |-  ( ( U  e.  Univ  /\  A  e. .Morphism  )  ->  (
( graph SetCat `  U ) `  A )  =  ( 2nd `  A ) )
8 fveq1 5524 . . . 4  |-  (.graph  =  ( graph SetCat `  U
)  ->  (.graph  `  A )  =  ( ( graph SetCat `  U ) `  A ) )
98eqeq1d 2291 . . 3  |-  (.graph  =  ( graph SetCat `  U
)  ->  ( (.graph  `  A )  =  ( 2nd `  A )  <-> 
( ( graph SetCat `  U
) `  A )  =  ( 2nd `  A
) ) )
107, 9syl5ibr 212 . 2  |-  (.graph  =  ( graph SetCat `  U
)  ->  ( ( U  e.  Univ  /\  A  e. .Morphism  )  ->  (.graph  `  A )  =  ( 2nd `  A ) ) )
111, 10ax-mp 8 1  |-  ( ( U  e.  Univ  /\  A  e. .Morphism  )  ->  (.graph  `  A )  =  ( 2nd `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   ` cfv 5255   2ndc2nd 6121   Univcgru 8412   Morphism SetCatccmrcase 25910   graph SetCatcgraphcase 25921
This theorem is referenced by:  grphidmor3  25954  cmp2morpcatt  25962  cmp2morpgrp  25963  morexcmp2  25968
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-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-pr 4214
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-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-graphcatset 25922
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