Theorem relfull 14922

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

Step	Hyp	Ref	Expression
1		fullfunc 14920	. 2 |- ( C Full D ) C_ ( C Func D )
2		relfunc 14876	. 2 |- Rel ( C Func D )
3		relss 5027	. 2 |- ( ( C Full D ) C_ ( C Func D ) -> ( Rel ( C Func D ) -> Rel ( C Full D ) ) )
4	1, 2, 3	mp2 9	1 |- Rel ( C Full D )

Syntax hints:   C_ wss 3428   Rel wrel 4945  (class class class)co 6192   Func cfunc 14868   Full cful 14916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-1st 6679  df-2nd 6680  df-func 14872  df-full 14918

This theorem is referenced by:  fullpropd  14934  cofull  14948