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Theorem monpropd 13963
 Description: If two categories have the same set of objects, morphisms, and compositions, then they have the same monomorphisms. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
monpropd.3 f f
monpropd.4 compf compf
monpropd.c
monpropd.d
Assertion
Ref Expression
monpropd Mono Mono

Proof of Theorem monpropd
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2436 . . . . . . . . . . . 12
2 eqid 2436 . . . . . . . . . . . 12
3 eqid 2436 . . . . . . . . . . . 12
4 monpropd.3 . . . . . . . . . . . . . 14 f f
54ad2antrr 707 . . . . . . . . . . . . 13 f f
65ad2antrr 707 . . . . . . . . . . . 12 f f
7 simpr 448 . . . . . . . . . . . 12
8 simp-4r 744 . . . . . . . . . . . 12
91, 2, 3, 6, 7, 8homfeqval 13923 . . . . . . . . . . 11
10 eqid 2436 . . . . . . . . . . . 12 comp comp
11 eqid 2436 . . . . . . . . . . . 12 comp comp
124ad5antr 715 . . . . . . . . . . . 12 f f
13 monpropd.4 . . . . . . . . . . . . 13 compf compf
1413ad5antr 715 . . . . . . . . . . . 12 compf compf
15 simplr 732 . . . . . . . . . . . 12
16 simp-5r 746 . . . . . . . . . . . 12
17 simp-4r 744 . . . . . . . . . . . 12
18 simpr 448 . . . . . . . . . . . 12
19 simpllr 736 . . . . . . . . . . . 12
201, 2, 10, 11, 12, 14, 15, 16, 17, 18, 19comfeqval 13934 . . . . . . . . . . 11 comp comp
219, 20mpteq12dva 4286 . . . . . . . . . 10 comp comp
2221cnveqd 5048 . . . . . . . . 9 comp comp
2322funeqd 5475 . . . . . . . 8 comp comp
2423ralbidva 2721 . . . . . . 7 comp comp
2524rabbidva 2947 . . . . . 6 comp comp
26 simplr 732 . . . . . . . 8
27 simpr 448 . . . . . . . 8
281, 2, 3, 5, 26, 27homfeqval 13923 . . . . . . 7
294homfeqbas 13922 . . . . . . . . 9
3029ad2antrr 707 . . . . . . . 8
3130raleqdv 2910 . . . . . . 7 comp comp
3228, 31rabeqbidv 2951 . . . . . 6 comp comp
3325, 32eqtrd 2468 . . . . 5 comp comp
34333impa 1148 . . . 4 comp comp
3534mpt2eq3dva 6138 . . 3 comp comp
36 mpt2eq12 6134 . . . 4 comp comp
3729, 29, 36syl2anc 643 . . 3 comp comp
3835, 37eqtrd 2468 . 2 comp comp
39 eqid 2436 . . 3 Mono Mono
40 monpropd.c . . 3
411, 2, 10, 39, 40monfval 13958 . 2 Mono comp
42 eqid 2436 . . 3
43 eqid 2436 . . 3 Mono Mono
44 monpropd.d . . 3
4542, 3, 11, 43, 44monfval 13958 . 2 Mono comp
4638, 41, 453eqtr4d 2478 1 Mono Mono
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725  wral 2705  crab 2709  cop 3817   cmpt 4266  ccnv 4877   wfun 5448  cfv 5454  (class class class)co 6081   cmpt2 6083  cbs 13469   chom 13540  compcco 13541  ccat 13889   f chomf 13891  compfccomf 13892  Monocmon 13954 This theorem is referenced by:  oppcepi  13965 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-homf 13895  df-comf 13896  df-mon 13956
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