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Theorem relfull 16772
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 16770 . 2 (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷)
2 relfunc 16726 . 2 Rel (𝐶 Func 𝐷)
3 relss 5408 . 2 ((𝐶 Full 𝐷) ⊆ (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:  wss 3769  Rel wrel 5316  (class class class)co 6874   Func cfunc 16718   Full cful 16766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6064  df-fun 6103  df-fv 6109  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-1st 7398  df-2nd 7399  df-func 16722  df-full 16768
This theorem is referenced by:  fullpropd  16784  cofull  16798
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