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Theorem fullfunc 14093
 Description: A full functor is a functor. (Contributed by Mario Carneiro, 26-Jan-2017.)
Assertion
Ref Expression
fullfunc Full

Proof of Theorem fullfunc
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 6080 . . . 4 Full Full
2 oveq1 6080 . . . 4
31, 2sseq12d 3369 . . 3 Full Full
4 oveq2 6081 . . . 4 Full Full
5 oveq2 6081 . . . 4
64, 5sseq12d 3369 . . 3 Full Full
7 ovex 6098 . . . . . 6
8 simpl 444 . . . . . . . 8
98ssopab2i 4474 . . . . . . 7
10 opabss 4261 . . . . . . 7
119, 10sstri 3349 . . . . . 6
127, 11ssexi 4340 . . . . 5
13 df-full 14091 . . . . . 6 Full
1413ovmpt4g 6188 . . . . 5 Full
1512, 14mp3an3 1268 . . . 4 Full
1615, 11syl6eqss 3390 . . 3 Full
173, 6, 16vtocl2ga 3011 . 2 Full
1813mpt2ndm0 6465 . . 3 Full
19 0ss 3648 . . 3
2018, 19syl6eqss 3390 . 2 Full
2117, 20pm2.61i 158 1 Full
 Colors of variables: wff set class Syntax hints:   wn 3   wa 359   wceq 1652   wcel 1725  wral 2697  cvv 2948   wss 3312  c0 3620   class class class wbr 4204  copab 4257   crn 4871  cfv 5446  (class class class)co 6073  cbs 13459   chom 13530  ccat 13879   cfunc 14041   Full cful 14089 This theorem is referenced by:  relfull  14095  isfull  14097  fulloppc  14109  cofull  14121  catcisolem  14251  catciso  14252 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-full 14091
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