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

Theorem relfull 15131
Description: The set of full functors is a relation. (Contributed by Mario Carneiro, 26-Jan-2017.)
Assertion
Ref Expression
relfull  |-  Rel  ( C Full  D )

Proof of Theorem relfull
StepHypRef Expression
1 fullfunc 15129 . 2  |-  ( C Full 
D )  C_  ( C  Func  D )
2 relfunc 15085 . 2  |-  Rel  ( C  Func  D )
3 relss 5088 . 2  |-  ( ( C Full  D )  C_  ( C  Func  D )  ->  ( Rel  ( C  Func  D )  ->  Rel  ( C Full  D ) ) )
41, 2, 3mp2 9 1  |-  Rel  ( C Full  D )
Colors of variables: wff setvar class
Syntax hints:    C_ wss 3476   Rel wrel 5004  (class class class)co 6282    Func cfunc 15077   Full cful 15125
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-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-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-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-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-func 15081  df-full 15127
This theorem is referenced by:  fullpropd  15143  cofull  15157
  Copyright terms: Public domain W3C validator