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Theorem cmpmorass 25966
 Description: Associativity of composition in category Set. (Contributed by FL, 7-Nov-2013.)
Hypotheses
Ref Expression
cmp2morpcatt.1
cmp2morpcatt.2 .Morphism
cmp2morpcatt.3 .dom
cmp2morpcatt.4 .cod
Assertion
Ref Expression
cmpmorass .Morphism .Morphism .Morphism .dom .cod .dom .cod

Proof of Theorem cmpmorass
StepHypRef Expression
1 simp1 955 . . 3 .Morphism .Morphism .Morphism .dom .cod .dom .cod
2 simp23 990 . . 3 .Morphism .Morphism .Morphism .dom .cod .dom .cod .Morphism
3 simp22 989 . . . 4 .Morphism .Morphism .Morphism .dom .cod .dom .cod .Morphism
4 simp21 988 . . . 4 .Morphism .Morphism .Morphism .dom .cod .dom .cod .Morphism
5 simp3r 984 . . . 4 .Morphism .Morphism .Morphism .dom .cod .dom .cod .dom .cod
6 cmp2morpcatt.1 . . . . 5
7 cmp2morpcatt.2 . . . . 5 .Morphism
8 cmp2morpcatt.3 . . . . 5 .dom
9 cmp2morpcatt.4 . . . . 5 .cod
106, 7, 8, 9rocatval2 25960 . . . 4 .Morphism .Morphism .dom .cod .Morphism
111, 3, 4, 5, 10syl121anc 1187 . . 3 .Morphism .Morphism .Morphism .dom .cod .dom .cod .Morphism
12 simp3l 983 . . . 4 .Morphism .Morphism .Morphism .dom .cod .dom .cod .dom .cod
136, 7, 8, 9cmp2morpcod 25965 . . . . 5 .Morphism .Morphism .dom .cod .cod .cod
141, 3, 4, 5, 13syl121anc 1187 . . . 4 .Morphism .Morphism .Morphism .dom .cod .dom .cod .cod .cod
1512, 14eqtr4d 2318 . . 3 .Morphism .Morphism .Morphism .dom .cod .dom .cod .dom .cod
16 eqid 2283 . . . 4
176, 7, 8, 9, 16cmp2morpcatt 25962 . . 3 .Morphism .Morphism .dom .cod .dom .cod
181, 2, 11, 15, 17syl121anc 1187 . 2 .Morphism .Morphism .Morphism .dom .cod .dom .cod .dom .cod
196, 7, 8, 9cmp2morpdom 25964 . . . . 5 .Morphism .Morphism .dom .cod .dom .dom
201, 3, 4, 5, 19syl121anc 1187 . . . 4 .Morphism .Morphism .Morphism .dom .cod .dom .cod .dom .dom
2120opeq1d 3802 . . 3 .Morphism .Morphism .Morphism .dom .cod .dom .cod .dom .cod .dom .cod
226, 7, 8, 9, 16cmp2morpgrp 25963 . . . . 5 .Morphism .Morphism .dom .cod
231, 3, 4, 5, 22syl121anc 1187 . . . 4 .Morphism .Morphism .Morphism .dom .cod .dom .cod
2423coeq2d 4846 . . 3 .Morphism .Morphism .Morphism .dom .cod .dom .cod
2521, 24opeq12d 3804 . 2 .Morphism .Morphism .Morphism .dom .cod .dom .cod .dom .cod .dom .cod
266, 7, 8, 9cmp2morpcod 25965 . . . . . . 7 .Morphism .Morphism .dom .cod .cod .cod
2726eqcomd 2288 . . . . . 6 .Morphism .Morphism .dom .cod .cod .cod
281, 2, 3, 12, 27syl121anc 1187 . . . . 5 .Morphism .Morphism .Morphism .dom .cod .dom .cod .cod .cod
2928opeq2d 3803 . . . 4 .Morphism .Morphism .Morphism .dom .cod .dom .cod .dom .cod .dom .cod
306, 7, 8, 9, 16cmp2morpgrp 25963 . . . . . . 7 .Morphism .Morphism .dom .cod
311, 2, 3, 12, 30syl121anc 1187 . . . . . 6 .Morphism .Morphism .Morphism .dom .cod .dom .cod
3231eqcomd 2288 . . . . 5 .Morphism .Morphism .Morphism .dom .cod .dom .cod
3332coeq1d 4845 . . . 4 .Morphism .Morphism .Morphism .dom .cod .dom .cod
3429, 33opeq12d 3804 . . 3 .Morphism .Morphism .Morphism .dom .cod .dom .cod .dom .cod .dom .cod
35 coass 5191 . . . . . 6
3635eqcomi 2287 . . . . 5
3736a1i 10 . . . 4 .Morphism .Morphism .Morphism .dom .cod .dom .cod
3837opeq2d 3803 . . 3 .Morphism .Morphism .Morphism .dom .cod .dom .cod .dom .cod .dom .cod
396, 7, 8, 9rocatval2 25960 . . . . 5 .Morphism .Morphism .dom .cod .Morphism
401, 2, 3, 12, 39syl121anc 1187 . . . 4 .Morphism .Morphism .Morphism .dom .cod .dom .cod .Morphism
416, 7, 8, 9cmp2morpdom 25964 . . . . . 6 .Morphism .Morphism .dom .cod .dom .dom
421, 2, 3, 12, 41syl121anc 1187 . . . . 5 .Morphism .Morphism .Morphism .dom .cod .dom .cod .dom .dom
4342, 5eqtrd 2315 . . . 4 .Morphism .Morphism .Morphism .dom .cod .dom .cod .dom .cod
446, 7, 8, 9, 16cmp2morpcatt 25962 . . . 4 .Morphism .Morphism .dom .cod .dom .cod
451, 40, 4, 43, 44syl121anc 1187 . . 3 .Morphism .Morphism .Morphism .dom .cod .dom .cod .dom .cod
4634, 38, 453eqtr4d 2325 . 2 .Morphism .Morphism .Morphism .dom .cod .dom .cod .dom .cod
4718, 25, 463eqtrd 2319 1 .Morphism .Morphism .Morphism .dom .cod .dom .cod
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   w3a 934   wceq 1623   wcel 1684  cop 3643   ccom 4693  cfv 5255  (class class class)co 5858  cgru 8412  ccmrcase 25910  cdomcase 25919  cgraphcase 25921  ccodcase 25932  crocase 25955 This theorem is referenced by:  setiscat  25979 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-morcatset 25911  df-domcatset 25920  df-graphcatset 25922  df-codcatset 25933  df-rocatset 25956
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