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Theorem homarel 13884
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 4809 . . . 4  |-  ( ( ( Base `  C
)  X.  ( Base `  C ) )  X. 
_V )  C_  ( _V  X.  _V )
2 homahom.h . . . . . . 7  |-  H  =  (Homa
`  C )
3 eqid 2296 . . . . . . 7  |-  ( Base `  C )  =  (
Base `  C )
42homarcl 13876 . . . . . . 7  |-  ( f  e.  ( X H Y )  ->  C  e.  Cat )
52, 3, 4homaf 13878 . . . . . 6  |-  ( f  e.  ( X H Y )  ->  H : ( ( Base `  C )  X.  ( Base `  C ) ) --> ~P ( ( (
Base `  C )  X.  ( Base `  C
) )  X.  _V ) )
62, 3homarcl2 13883 . . . . . . 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 6008 . . . . 5  |-  ( f  e.  ( X H Y )  ->  ( X H Y )  e. 
~P ( ( (
Base `  C )  X.  ( Base `  C
) )  X.  _V ) )
10 elelpwi 3648 . . . . 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 3191 . . 3  |-  ( f  e.  ( X H Y )  ->  f  e.  ( _V  X.  _V ) )
1312ssriv 3197 . 2  |-  ( X H Y )  C_  ( _V  X.  _V )
14 df-rel 4712 . 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 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638    X. cxp 4703   Rel wrel 4710   ` cfv 5271  (class class class)co 5874   Basecbs 13164  Homachoma 13871
This theorem is referenced by:  homahom  13887  homadm  13888  homacd  13889  homadmcd  13890
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-ov 5877  df-homa 13874
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