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Theorem fullpropd 14118
 Description: If two categories have the same set of objects, morphisms, and compositions, then they have the same full functors. (Contributed by Mario Carneiro, 27-Jan-2017.)
Hypotheses
Ref Expression
fullpropd.1 f f
fullpropd.2 compf compf
fullpropd.3 f f
fullpropd.4 compf compf
fullpropd.a
fullpropd.b
fullpropd.c
fullpropd.d
Assertion
Ref Expression
fullpropd Full Full

Proof of Theorem fullpropd
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relfull 14106 . 2 Full
2 relfull 14106 . 2 Full
3 fullpropd.1 . . . . . . . 8 f f
43homfeqbas 13923 . . . . . . 7
54adantr 453 . . . . . 6
65adantr 453 . . . . . . 7
7 eqid 2437 . . . . . . . . 9
8 eqid 2437 . . . . . . . . 9
9 eqid 2437 . . . . . . . . 9
10 fullpropd.3 . . . . . . . . . 10 f f
1110ad3antrrr 712 . . . . . . . . 9 f f
12 eqid 2437 . . . . . . . . . . 11
13 simpllr 737 . . . . . . . . . . 11
1412, 7, 13funcf1 14064 . . . . . . . . . 10
15 simplr 733 . . . . . . . . . 10
1614, 15ffvelrnd 5872 . . . . . . . . 9
17 simpr 449 . . . . . . . . . 10
1814, 17ffvelrnd 5872 . . . . . . . . 9
197, 8, 9, 11, 16, 18homfeqval 13924 . . . . . . . 8
2019eqeq2d 2448 . . . . . . 7
216, 20raleqbidva 2919 . . . . . 6
225, 21raleqbidva 2919 . . . . 5
2322pm5.32da 624 . . . 4
24 fullpropd.2 . . . . . . 7 compf compf
25 fullpropd.4 . . . . . . 7 compf compf
26 fullpropd.a . . . . . . 7
27 fullpropd.b . . . . . . 7
28 fullpropd.c . . . . . . 7
29 fullpropd.d . . . . . . 7
303, 24, 10, 25, 26, 27, 28, 29funcpropd 14098 . . . . . 6
3130breqd 4224 . . . . 5
3231anbi1d 687 . . . 4
3323, 32bitrd 246 . . 3
3412, 8isfull 14108 . . 3 Full
35 eqid 2437 . . . 4
3635, 9isfull 14108 . . 3 Full
3733, 34, 363bitr4g 281 . 2 Full Full
381, 2, 37eqbrrdiv 4975 1 Full Full
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   wceq 1653   wcel 1726  wral 2706   class class class wbr 4213   crn 4880  cfv 5455  (class class class)co 6082  cbs 13470   chom 13541   f chomf 13892  compfccomf 13893   cfunc 14052   Full cful 14100 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-map 7021  df-ixp 7065  df-cat 13894  df-cid 13895  df-homf 13896  df-comf 13897  df-func 14056  df-full 14102
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