Theorem relfull 15806

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 15804	. 2 |- ( C Full D ) C_ ( C Func D )
2		relfunc 15760	. 2 |- Rel ( C Func D )
3		relss 4939	. 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 3437   Rel wrel 4856  (class class class)co 6303   Func cfunc 15752   Full cful 15800

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fv 5607  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-1st 6805  df-2nd 6806  df-func 15756  df-full 15802

This theorem is referenced by:  fullpropd  15818  cofull  15832