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Theorem fthmon 14129
 Description: A faithful functor reflects monomorphisms. (Contributed by Mario Carneiro, 27-Jan-2017.)
Hypotheses
Ref Expression
fthmon.b
fthmon.h
fthmon.f Faith
fthmon.x
fthmon.y
fthmon.r
fthmon.m Mono
fthmon.n Mono
fthmon.1
Assertion
Ref Expression
fthmon

Proof of Theorem fthmon
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fthmon.r . 2
2 eqid 2438 . . . . . 6
3 eqid 2438 . . . . . 6
4 eqid 2438 . . . . . 6 comp comp
5 fthmon.n . . . . . 6 Mono
6 fthmon.f . . . . . . . . . . 11 Faith
7 fthfunc 14109 . . . . . . . . . . . 12 Faith
87ssbri 4257 . . . . . . . . . . 11 Faith
96, 8syl 16 . . . . . . . . . 10
10 df-br 4216 . . . . . . . . . 10
119, 10sylib 190 . . . . . . . . 9
12 funcrcl 14065 . . . . . . . . 9
1311, 12syl 16 . . . . . . . 8
1413simprd 451 . . . . . . 7
1514adantr 453 . . . . . 6
16 fthmon.b . . . . . . . 8
179adantr 453 . . . . . . . 8
1816, 2, 17funcf1 14068 . . . . . . 7
19 fthmon.x . . . . . . . 8
2019adantr 453 . . . . . . 7
2118, 20ffvelrnd 5874 . . . . . 6
22 fthmon.y . . . . . . . 8
2322adantr 453 . . . . . . 7
2418, 23ffvelrnd 5874 . . . . . 6
25 simpr1 964 . . . . . . 7
2618, 25ffvelrnd 5874 . . . . . 6
27 fthmon.1 . . . . . . 7
2827adantr 453 . . . . . 6
29 fthmon.h . . . . . . . 8
3016, 29, 3, 17, 25, 20funcf2 14070 . . . . . . 7
31 simpr2 965 . . . . . . 7
3230, 31ffvelrnd 5874 . . . . . 6
33 simpr3 966 . . . . . . 7
3430, 33ffvelrnd 5874 . . . . . 6
352, 3, 4, 5, 15, 21, 24, 26, 28, 32, 34moni 13967 . . . . 5 comp comp
36 eqid 2438 . . . . . . . 8 comp comp
371adantr 453 . . . . . . . 8
3816, 29, 36, 4, 17, 25, 20, 23, 31, 37funcco 14073 . . . . . . 7 comp comp
3916, 29, 36, 4, 17, 25, 20, 23, 33, 37funcco 14073 . . . . . . 7 comp comp
4038, 39eqeq12d 2452 . . . . . 6 comp comp comp comp
416adantr 453 . . . . . . 7 Faith
4213simpld 447 . . . . . . . . 9
4342adantr 453 . . . . . . . 8
4416, 29, 36, 43, 25, 20, 23, 31, 37catcocl 13915 . . . . . . 7 comp
4516, 29, 36, 43, 25, 20, 23, 33, 37catcocl 13915 . . . . . . 7 comp
4616, 29, 3, 41, 25, 23, 44, 45fthi 14120 . . . . . 6 comp comp comp comp
4740, 46bitr3d 248 . . . . 5 comp comp comp comp
4816, 29, 3, 41, 25, 20, 31, 33fthi 14120 . . . . 5
4935, 47, 483bitr3d 276 . . . 4 comp comp
5049biimpd 200 . . 3 comp comp
5150ralrimivvva 2801 . 2 comp comp
52 fthmon.m . . 3 Mono
5316, 29, 36, 52, 42, 19, 22ismon2 13965 . 2 comp comp
541, 51, 53mpbir2and 890 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   w3a 937   wceq 1653   wcel 1726  wral 2707  cop 3819   class class class wbr 4215  cfv 5457  (class class class)co 6084  cbs 13474   chom 13545  compcco 13546  ccat 13894  Monocmon 13959   cfunc 14056   Faith cfth 14105 This theorem is referenced by:  fthepi  14130 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704 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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-map 7023  df-ixp 7067  df-cat 13898  df-mon 13961  df-func 14060  df-fth 14107
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