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Theorem homarel 13868
Description: An arrow is an ordered pair. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypothesis
Ref Expression
homahom.h  |-  H  =  (Homa
`  C )
Assertion
Ref Expression
homarel  |-  Rel  ( X H Y )

Proof of Theorem homarel
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 xpss 4793 . . . 4  |-  ( ( ( Base `  C
)  X.  ( Base `  C ) )  X. 
_V )  C_  ( _V  X.  _V )
2 homahom.h . . . . . . 7  |-  H  =  (Homa
`  C )
3 eqid 2283 . . . . . . 7  |-  ( Base `  C )  =  (
Base `  C )
42homarcl 13860 . . . . . . 7  |-  ( f  e.  ( X H Y )  ->  C  e.  Cat )
52, 3, 4homaf 13862 . . . . . 6  |-  ( f  e.  ( X H Y )  ->  H : ( ( Base `  C )  X.  ( Base `  C ) ) --> ~P ( ( (
Base `  C )  X.  ( Base `  C
) )  X.  _V ) )
62, 3homarcl2 13867 . . . . . . 7  |-  ( f  e.  ( X H Y )  ->  ( X  e.  ( Base `  C )  /\  Y  e.  ( Base `  C
) ) )
76simpld 445 . . . . . 6  |-  ( f  e.  ( X H Y )  ->  X  e.  ( Base `  C
) )
86simprd 449 . . . . . 6  |-  ( f  e.  ( X H Y )  ->  Y  e.  ( Base `  C
) )
95, 7, 8fovrnd 5992 . . . . 5  |-  ( f  e.  ( X H Y )  ->  ( X H Y )  e. 
~P ( ( (
Base `  C )  X.  ( Base `  C
) )  X.  _V ) )
10 elelpwi 3635 . . . . 5  |-  ( ( f  e.  ( X H Y )  /\  ( X H Y )  e.  ~P ( ( ( Base `  C
)  X.  ( Base `  C ) )  X. 
_V ) )  -> 
f  e.  ( ( ( Base `  C
)  X.  ( Base `  C ) )  X. 
_V ) )
119, 10mpdan 649 . . . 4  |-  ( f  e.  ( X H Y )  ->  f  e.  ( ( ( Base `  C )  X.  ( Base `  C ) )  X.  _V ) )
121, 11sseldi 3178 . . 3  |-  ( f  e.  ( X H Y )  ->  f  e.  ( _V  X.  _V ) )
1312ssriv 3184 . 2  |-  ( X H Y )  C_  ( _V  X.  _V )
14 df-rel 4696 . 2  |-  ( Rel  ( X H Y )  <->  ( X H Y )  C_  ( _V  X.  _V ) )
1513, 14mpbir 200 1  |-  Rel  ( X H Y )
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625    X. cxp 4687   Rel wrel 4694   ` cfv 5255  (class class class)co 5858   Basecbs 13148  Homachoma 13855
This theorem is referenced by:  homahom  13871  homadm  13872  homacd  13873  homadmcd  13874
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-homa 13858
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