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Theorem fullfunc 13796
 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 5881 . . . 4 Full Full
2 oveq1 5881 . . . 4
31, 2sseq12d 3220 . . 3 Full Full
4 oveq2 5882 . . . 4 Full Full
5 oveq2 5882 . . . 4
64, 5sseq12d 3220 . . 3 Full Full
7 ovex 5899 . . . . . 6
8 simpl 443 . . . . . . . 8
98ssopab2i 4308 . . . . . . 7
10 opabss 4096 . . . . . . 7
119, 10sstri 3201 . . . . . 6
127, 11ssexi 4175 . . . . 5
13 df-full 13794 . . . . . 6 Full
1413ovmpt4g 5986 . . . . 5 Full
1512, 14mp3an3 1266 . . . 4 Full
1615, 11syl6eqss 3241 . . 3 Full
173, 6, 16vtocl2ga 2864 . 2 Full
1813, 12fnmpt2i 6209 . . . . 5 Full
19 fndm 5359 . . . . 5 Full Full
2018, 19ax-mp 8 . . . 4 Full
2120ndmov 6020 . . 3 Full
22 0ss 3496 . . 3
2321, 22syl6eqss 3241 . 2 Full
2417, 23pm2.61i 156 1 Full
 Colors of variables: wff set class Syntax hints:   wn 3   wa 358   wceq 1632   wcel 1696  wral 2556  cvv 2801   wss 3165  c0 3468   class class class wbr 4039  copab 4092   cxp 4703   cdm 4705   crn 4706   wfn 5266  cfv 5271  (class class class)co 5874  cbs 13164   chom 13235  ccat 13582   cfunc 13744   Full cful 13792 This theorem is referenced by:  relfull  13798  isfull  13800  fulloppc  13812  cofull  13824  catcisolem  13954  catciso  13955 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-full 13794
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