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Theorem sectmon 15044
 Description: If is a section of , then is a monomorphism. A monomorphism that arises from a section is also known as a split monomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
sectmon.b
sectmon.m Mono
sectmon.s Sect
sectmon.c
sectmon.x
sectmon.y
sectmon.1
Assertion
Ref Expression
sectmon

Proof of Theorem sectmon
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sectmon.1 . . . 4
2 sectmon.b . . . . 5
3 eqid 2441 . . . . 5
4 eqid 2441 . . . . 5 comp comp
5 eqid 2441 . . . . 5
6 sectmon.s . . . . 5 Sect
7 sectmon.c . . . . 5
8 sectmon.x . . . . 5
9 sectmon.y . . . . 5
102, 3, 4, 5, 6, 7, 8, 9issect 15020 . . . 4 comp
111, 10mpbid 210 . . 3 comp
1211simp1d 1007 . 2
13 oveq2 6285 . . . . 5 comp comp comp comp comp comp
1411simp3d 1009 . . . . . . . . 9 comp
1514ad2antrr 725 . . . . . . . 8 comp
1615oveq1d 6292 . . . . . . 7 comp comp comp
177ad2antrr 725 . . . . . . . 8
18 simplr 754 . . . . . . . 8
198ad2antrr 725 . . . . . . . 8
209ad2antrr 725 . . . . . . . 8
21 simprl 755 . . . . . . . 8
2212ad2antrr 725 . . . . . . . 8
2311simp2d 1008 . . . . . . . . 9
2423ad2antrr 725 . . . . . . . 8
252, 3, 4, 17, 18, 19, 20, 21, 22, 19, 24catass 14955 . . . . . . 7 comp comp comp comp
262, 3, 5, 17, 18, 4, 19, 21catlid 14952 . . . . . . 7 comp
2716, 25, 263eqtr3d 2490 . . . . . 6 comp comp
2815oveq1d 6292 . . . . . . 7 comp comp comp
29 simprr 756 . . . . . . . 8
302, 3, 4, 17, 18, 19, 20, 29, 22, 19, 24catass 14955 . . . . . . 7 comp comp comp comp
312, 3, 5, 17, 18, 4, 19, 29catlid 14952 . . . . . . 7 comp
3228, 30, 313eqtr3d 2490 . . . . . 6 comp comp
3327, 32eqeq12d 2463 . . . . 5 comp comp comp comp
3413, 33syl5ib 219 . . . 4 comp comp
3534ralrimivva 2862 . . 3 comp comp
3635ralrimiva 2855 . 2 comp comp
37 sectmon.m . . 3 Mono
382, 3, 4, 37, 7, 8, 9ismon2 15001 . 2 comp comp
3912, 36, 38mpbir2and 920 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   w3a 972   wceq 1381   wcel 1802  wral 2791  cop 4016   class class class wbr 4433  cfv 5574  (class class class)co 6277  cbs 14504   chom 14580  compcco 14581  ccat 14933  ccid 14934  Monocmon 14995  Sectcsect 15011 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6781  df-2nd 6782  df-cat 14937  df-cid 14938  df-mon 14997  df-sect 15014 This theorem is referenced by:  sectepi  15046
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