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Theorem cofull 14059
 Description: The composition of two full functors is full. (Contributed by Mario Carneiro, 28-Jan-2017.)
Hypotheses
Ref Expression
cofull.f Full
cofull.g Full
Assertion
Ref Expression
cofull func Full

Proof of Theorem cofull
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relfunc 13987 . . 3
2 fullfunc 14031 . . . . 5 Full
3 cofull.f . . . . 5 Full
42, 3sseldi 3290 . . . 4
5 fullfunc 14031 . . . . 5 Full
6 cofull.g . . . . 5 Full
75, 6sseldi 3290 . . . 4
84, 7cofucl 14013 . . 3 func
9 1st2nd 6333 . . 3 func func func func
101, 8, 9sylancr 645 . 2 func func func
11 1st2ndbr 6336 . . . . 5 func func func
121, 8, 11sylancr 645 . . . 4 func func
13 eqid 2388 . . . . . . . 8
14 eqid 2388 . . . . . . . 8
15 eqid 2388 . . . . . . . 8
16 relfull 14033 . . . . . . . . 9 Full
176adantr 452 . . . . . . . . 9 Full
18 1st2ndbr 6336 . . . . . . . . 9 Full Full Full
1916, 17, 18sylancr 645 . . . . . . . 8 Full
20 eqid 2388 . . . . . . . . . 10
21 relfunc 13987 . . . . . . . . . . 11
224adantr 452 . . . . . . . . . . 11
23 1st2ndbr 6336 . . . . . . . . . . 11
2421, 22, 23sylancr 645 . . . . . . . . . 10
2520, 13, 24funcf1 13991 . . . . . . . . 9
26 simprl 733 . . . . . . . . 9
2725, 26ffvelrnd 5811 . . . . . . . 8
28 simprr 734 . . . . . . . . 9
2925, 28ffvelrnd 5811 . . . . . . . 8
3013, 14, 15, 19, 27, 29fullfo 14037 . . . . . . 7
31 eqid 2388 . . . . . . . 8
32 relfull 14033 . . . . . . . . 9 Full
333adantr 452 . . . . . . . . 9 Full
34 1st2ndbr 6336 . . . . . . . . 9 Full Full Full
3532, 33, 34sylancr 645 . . . . . . . 8 Full
3620, 15, 31, 35, 26, 28fullfo 14037 . . . . . . 7
37 foco 5604 . . . . . . 7
3830, 36, 37syl2anc 643 . . . . . 6
397adantr 452 . . . . . . . 8
4020, 22, 39, 26, 28cofu2nd 14010 . . . . . . 7 func
41 eqidd 2389 . . . . . . 7
4220, 22, 39, 26cofu1 14009 . . . . . . . 8 func
4320, 22, 39, 28cofu1 14009 . . . . . . . 8 func
4442, 43oveq12d 6039 . . . . . . 7 func func
4540, 41, 44foeq123d 5611 . . . . . 6 func func func
4638, 45mpbird 224 . . . . 5 func func func
4746ralrimivva 2742 . . . 4 func func func
4820, 14, 31isfull2 14036 . . . 4 func Full func func func func func func
4912, 47, 48sylanbrc 646 . . 3 func Full func
50 df-br 4155 . . 3 func Full func func func Full
5149, 50sylib 189 . 2 func func Full
5210, 51eqeltrd 2462 1 func Full
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1649   wcel 1717  wral 2650  cop 3761   class class class wbr 4154   ccom 4823   wrel 4824  wfo 5393  cfv 5395  (class class class)co 6021  c1st 6287  c2nd 6288  cbs 13397   chom 13468   cfunc 13979   func ccofu 13981   Full cful 14027 This theorem is referenced by:  coffth  14061 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-map 6957  df-ixp 7001  df-cat 13821  df-cid 13822  df-func 13983  df-cofu 13985  df-full 14029
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