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Theorem monsect 14005
 Description: If is a monomorphism and is a section of , then is an inverse of and they are both isomorphisms. This is also stated as "a monomorphism which is also a split epimorphism is an isomorphism". (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
sectmon.b
sectmon.m Mono
sectmon.s Sect
sectmon.c
sectmon.x
sectmon.y
monsect.n Inv
monsect.1
monsect.2
Assertion
Ref Expression
monsect

Proof of Theorem monsect
StepHypRef Expression
1 monsect.2 . . . . . . . 8
2 sectmon.b . . . . . . . . 9
3 eqid 2437 . . . . . . . . 9
4 eqid 2437 . . . . . . . . 9 comp comp
5 eqid 2437 . . . . . . . . 9
6 sectmon.s . . . . . . . . 9 Sect
7 sectmon.c . . . . . . . . 9
8 sectmon.y . . . . . . . . 9
9 sectmon.x . . . . . . . . 9
102, 3, 4, 5, 6, 7, 8, 9issect 13980 . . . . . . . 8 comp
111, 10mpbid 203 . . . . . . 7 comp
1211simp3d 972 . . . . . 6 comp
1312oveq1d 6097 . . . . 5 comp comp comp
1411simp2d 971 . . . . . 6
1511simp1d 970 . . . . . 6
162, 3, 4, 7, 9, 8, 9, 14, 15, 8, 14catass 13912 . . . . 5 comp comp comp comp
172, 3, 5, 7, 9, 4, 8, 14catlid 13909 . . . . . 6 comp
182, 3, 5, 7, 9, 4, 8, 14catrid 13910 . . . . . 6 comp
1917, 18eqtr4d 2472 . . . . 5 comp comp
2013, 16, 193eqtr3d 2477 . . . 4 comp comp comp
21 sectmon.m . . . . 5 Mono
22 monsect.1 . . . . 5
232, 3, 4, 7, 9, 8, 9, 14, 15catcocl 13911 . . . . 5 comp
242, 3, 5, 7, 9catidcl 13908 . . . . 5
252, 3, 4, 21, 7, 9, 8, 9, 22, 23, 24moni 13963 . . . 4 comp comp comp comp
2620, 25mpbid 203 . . 3 comp
272, 3, 4, 5, 6, 7, 9, 8, 14, 15issect2 13981 . . 3 comp
2826, 27mpbird 225 . 2
29 monsect.n . . 3 Inv
302, 29, 7, 9, 8, 6isinv 13986 . 2
3128, 1, 30mpbir2and 890 1
 Colors of variables: wff set class Syntax hints:   wi 4   w3a 937   wceq 1653   wcel 1726  cop 3818   class class class wbr 4213  cfv 5455  (class class class)co 6082  cbs 13470   chom 13541  compcco 13542  ccat 13890  ccid 13891  Monocmon 13955  Sectcsect 13971  Invcinv 13972 This theorem is referenced by:  episect  14007 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-rmo 2714  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-cat 13894  df-cid 13895  df-mon 13957  df-sect 13974  df-inv 13975
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