MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  homarel Structured version   Unicode version

Theorem homarel 15641
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 4932 . . . 4  |-  ( ( ( Base `  C
)  X.  ( Base `  C ) )  X. 
_V )  C_  ( _V  X.  _V )
2 homahom.h . . . . . . 7  |-  H  =  (Homa
`  C )
3 eqid 2404 . . . . . . 7  |-  ( Base `  C )  =  (
Base `  C )
42homarcl 15633 . . . . . . 7  |-  ( f  e.  ( X H Y )  ->  C  e.  Cat )
52, 3, 4homaf 15635 . . . . . 6  |-  ( f  e.  ( X H Y )  ->  H : ( ( Base `  C )  X.  ( Base `  C ) ) --> ~P ( ( (
Base `  C )  X.  ( Base `  C
) )  X.  _V ) )
62, 3homarcl2 15640 . . . . . . 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 6430 . . . . 5  |-  ( f  e.  ( X H Y )  ->  ( X H Y )  e. 
~P ( ( (
Base `  C )  X.  ( Base `  C
) )  X.  _V ) )
10 elelpwi 3968 . . . . 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 3442 . . 3  |-  ( f  e.  ( X H Y )  ->  f  e.  ( _V  X.  _V ) )
1312ssriv 3448 . 2  |-  ( X H Y )  C_  ( _V  X.  _V )
14 df-rel 4832 . 2  |-  ( Rel  ( X H Y )  <->  ( X H Y )  C_  ( _V  X.  _V ) )
1513, 14mpbir 211 1  |-  Rel  ( X H Y )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1407    e. wcel 1844   _Vcvv 3061    C_ wss 3416   ~Pcpw 3957    X. cxp 4823   Rel wrel 4830   ` cfv 5571  (class class class)co 6280   Basecbs 14843  Homachoma 15628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-ov 6283  df-homa 15631
This theorem is referenced by:  homahom  15644  homadm  15645  homacd  15646  homadmcd  15647
  Copyright terms: Public domain W3C validator