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Theorem homarel 14119
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 4923 . . . 4  |-  ( ( ( Base `  C
)  X.  ( Base `  C ) )  X. 
_V )  C_  ( _V  X.  _V )
2 homahom.h . . . . . . 7  |-  H  =  (Homa
`  C )
3 eqid 2388 . . . . . . 7  |-  ( Base `  C )  =  (
Base `  C )
42homarcl 14111 . . . . . . 7  |-  ( f  e.  ( X H Y )  ->  C  e.  Cat )
52, 3, 4homaf 14113 . . . . . 6  |-  ( f  e.  ( X H Y )  ->  H : ( ( Base `  C )  X.  ( Base `  C ) ) --> ~P ( ( (
Base `  C )  X.  ( Base `  C
) )  X.  _V ) )
62, 3homarcl2 14118 . . . . . . 7  |-  ( f  e.  ( X H Y )  ->  ( X  e.  ( Base `  C )  /\  Y  e.  ( Base `  C
) ) )
76simpld 446 . . . . . 6  |-  ( f  e.  ( X H Y )  ->  X  e.  ( Base `  C
) )
86simprd 450 . . . . . 6  |-  ( f  e.  ( X H Y )  ->  Y  e.  ( Base `  C
) )
95, 7, 8fovrnd 6158 . . . . 5  |-  ( f  e.  ( X H Y )  ->  ( X H Y )  e. 
~P ( ( (
Base `  C )  X.  ( Base `  C
) )  X.  _V ) )
10 elelpwi 3753 . . . . 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 650 . . . 4  |-  ( f  e.  ( X H Y )  ->  f  e.  ( ( ( Base `  C )  X.  ( Base `  C ) )  X.  _V ) )
121, 11sseldi 3290 . . 3  |-  ( f  e.  ( X H Y )  ->  f  e.  ( _V  X.  _V ) )
1312ssriv 3296 . 2  |-  ( X H Y )  C_  ( _V  X.  _V )
14 df-rel 4826 . 2  |-  ( Rel  ( X H Y )  <->  ( X H Y )  C_  ( _V  X.  _V ) )
1513, 14mpbir 201 1  |-  Rel  ( X H Y )
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1717   _Vcvv 2900    C_ wss 3264   ~Pcpw 3743    X. cxp 4817   Rel wrel 4824   ` cfv 5395  (class class class)co 6021   Basecbs 13397  Homachoma 14106
This theorem is referenced by:  homahom  14122  homadm  14123  homacd  14124  homadmcd  14125
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-homa 14109
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