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Theorem homarel 15217
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 5107 . . . 4  |-  ( ( ( Base `  C
)  X.  ( Base `  C ) )  X. 
_V )  C_  ( _V  X.  _V )
2 homahom.h . . . . . . 7  |-  H  =  (Homa
`  C )
3 eqid 2467 . . . . . . 7  |-  ( Base `  C )  =  (
Base `  C )
42homarcl 15209 . . . . . . 7  |-  ( f  e.  ( X H Y )  ->  C  e.  Cat )
52, 3, 4homaf 15211 . . . . . 6  |-  ( f  e.  ( X H Y )  ->  H : ( ( Base `  C )  X.  ( Base `  C ) ) --> ~P ( ( (
Base `  C )  X.  ( Base `  C
) )  X.  _V ) )
62, 3homarcl2 15216 . . . . . . 7  |-  ( f  e.  ( X H Y )  ->  ( X  e.  ( Base `  C )  /\  Y  e.  ( Base `  C
) ) )
76simpld 459 . . . . . 6  |-  ( f  e.  ( X H Y )  ->  X  e.  ( Base `  C
) )
86simprd 463 . . . . . 6  |-  ( f  e.  ( X H Y )  ->  Y  e.  ( Base `  C
) )
95, 7, 8fovrnd 6429 . . . . 5  |-  ( f  e.  ( X H Y )  ->  ( X H Y )  e. 
~P ( ( (
Base `  C )  X.  ( Base `  C
) )  X.  _V ) )
10 elelpwi 4021 . . . . 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 668 . . . 4  |-  ( f  e.  ( X H Y )  ->  f  e.  ( ( ( Base `  C )  X.  ( Base `  C ) )  X.  _V ) )
121, 11sseldi 3502 . . 3  |-  ( f  e.  ( X H Y )  ->  f  e.  ( _V  X.  _V ) )
1312ssriv 3508 . 2  |-  ( X H Y )  C_  ( _V  X.  _V )
14 df-rel 5006 . 2  |-  ( Rel  ( X H Y )  <->  ( X H Y )  C_  ( _V  X.  _V ) )
1513, 14mpbir 209 1  |-  Rel  ( X H Y )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    e. wcel 1767   _Vcvv 3113    C_ wss 3476   ~Pcpw 4010    X. cxp 4997   Rel wrel 5004   ` cfv 5586  (class class class)co 6282   Basecbs 14486  Homachoma 15204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-homa 15207
This theorem is referenced by:  homahom  15220  homadm  15221  homacd  15222  homadmcd  15223
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