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Theorem fullpropd 14826
 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 14814 . 2 Full
2 relfull 14814 . 2 Full
3 fullpropd.1 . . . . . . . 8 f f
43homfeqbas 14631 . . . . . . 7
54adantr 462 . . . . . 6
65adantr 462 . . . . . . 7
7 eqid 2441 . . . . . . . . 9
8 eqid 2441 . . . . . . . . 9
9 eqid 2441 . . . . . . . . 9
10 fullpropd.3 . . . . . . . . . 10 f f
1110ad3antrrr 724 . . . . . . . . 9 f f
12 eqid 2441 . . . . . . . . . . 11
13 simpllr 753 . . . . . . . . . . 11
1412, 7, 13funcf1 14772 . . . . . . . . . 10
15 simplr 749 . . . . . . . . . 10
1614, 15ffvelrnd 5841 . . . . . . . . 9
17 simpr 458 . . . . . . . . . 10
1814, 17ffvelrnd 5841 . . . . . . . . 9
197, 8, 9, 11, 16, 18homfeqval 14632 . . . . . . . 8
2019eqeq2d 2452 . . . . . . 7
216, 20raleqbidva 2931 . . . . . 6
225, 21raleqbidva 2931 . . . . 5
2322pm5.32da 636 . . . 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 14806 . . . . . 6
3130breqd 4300 . . . . 5
3231anbi1d 699 . . . 4
3323, 32bitrd 253 . . 3
3412, 8isfull 14816 . . 3 Full
35 eqid 2441 . . . 4
3635, 9isfull 14816 . . 3 Full
3733, 34, 363bitr4g 288 . 2 Full Full
381, 2, 37eqbrrdiv 4934 1 Full Full
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wceq 1364   wcel 1761  wral 2713   class class class wbr 4289   crn 4837  cfv 5415  (class class class)co 6090  cbs 14170   chom 14245   f chomf 14600  compfccomf 14601   cfunc 14760   Full cful 14808 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-map 7212  df-ixp 7260  df-cat 14602  df-cid 14603  df-homf 14604  df-comf 14605  df-func 14764  df-full 14810 This theorem is referenced by: (None)
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