Theorem relfull 17613

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 17611	. 2 ⊢ (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷)
2		relfunc 17566	. 2 ⊢ Rel (𝐶 Func 𝐷)
3		relss 5688	. 2 ⊢ ((𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷) → (Rel (𝐶 Func 𝐷) → Rel (𝐶 Full 𝐷)))
4	1, 2, 3	mp2 9	1 ⊢ Rel (𝐶 Full 𝐷)

Syntax hints:   ⊆ wss 3888  Rel wrel 5591  (class class class)co 7269   Func cfunc 17558   Full cful 17607

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7580

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3433  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-if 4462  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-iun 4928  df-br 5076  df-opab 5138  df-mpt 5159  df-id 5486  df-xp 5592  df-rel 5593  df-cnv 5594  df-co 5595  df-dm 5596  df-rn 5597  df-res 5598  df-ima 5599  df-iota 6386  df-fun 6430  df-fv 6436  df-ov 7272  df-oprab 7273  df-mpo 7274  df-1st 7822  df-2nd 7823  df-func 17562  df-full 17609

This theorem is referenced by:  fullpropd  17625  cofull  17639  thincciso  46287