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Theorem cmpidmor2 26072
 Description: Composition with an identity. (Contributed by FL, 8-Nov-2013.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
Hypotheses
Ref Expression
cmpidmor2.1
cmpidmor2.2 .Morphism
cmpidmor2.4 .cod
cmpidmor2.5 .id
Assertion
Ref Expression
cmpidmor2 .Morphism .id .cod

Proof of Theorem cmpidmor2
StepHypRef Expression
1 cmpidmor2.4 . . . . . 6 .cod
2 cmpidmor2.5 . . . . . . 7 .id
32fveq1i 5542 . . . . . 6 .id .cod .cod
41, 3fveq12i 5546 . . . . 5 .cod .id .cod .cod
51fveq1i 5542 . . . . . . 7 .cod
6 cmpidmor2.2 . . . . . . . . 9 .Morphism
76eleq2i 2360 . . . . . . . 8 .Morphism
8 codcatsetval 26038 . . . . . . . 8
97, 8sylan2b 461 . . . . . . 7 .Morphism
105, 9syl5eqel 2380 . . . . . 6 .Morphism .cod
11 codidmor 26053 . . . . . 6 .cod .cod .cod
1210, 11syldan 456 . . . . 5 .Morphism .cod .cod
134, 12syl5eq 2340 . . . 4 .Morphism .cod .id .cod .cod
1413opeq2d 3819 . . 3 .Morphism .cod .id .cod .cod
153fveq2i 5544 . . . . . 6 .id .cod .cod
16 eqid 2296 . . . . . . . 8
17 eqid 2296 . . . . . . . 8
1816, 17grphidmor2 26056 . . . . . . 7 .cod .cod .cod
1910, 18syldan 456 . . . . . 6 .Morphism .cod .cod
2015, 19syl5eq 2340 . . . . 5 .Morphism .id .cod .cod
2120coeq1d 4861 . . . 4 .Morphism .id .cod .cod
22 isgraphmrph 26026 . . . . . . . 8
237, 22sylan2b 461 . . . . . . 7 .Morphism
24 eqid 2296 . . . . . . . 8
25 eqid 2296 . . . . . . . 8
2624, 1, 25, 6prismorcset3 26041 . . . . . . 7 .Morphism .cod
2723, 26eqeltrrd 2371 . . . . . 6 .Morphism .cod
28 fvex 5555 . . . . . . 7 .cod
29 fvex 5555 . . . . . . 7
3028, 29elmap 6812 . . . . . 6 .cod .cod
3127, 30sylib 188 . . . . 5 .Morphism .cod
32 fcoi2 5432 . . . . 5 .cod .cod
3331, 32syl 15 . . . 4 .Morphism .cod
3421, 33eqtrd 2328 . . 3 .Morphism .id .cod
3514, 34opeq12d 3820 . 2 .Morphism .cod .id .cod .id .cod .cod
36 simpl 443 . . 3 .Morphism
372, 6idcatval2 26047 . . . 4 .cod .id .cod .Morphism
3810, 37syldan 456 . . 3 .Morphism .id .cod .Morphism
39 simpr 447 . . 3 .Morphism .Morphism
403fveq2i 5544 . . . . 5 .id .cod .cod
41 domidmor 26051 . . . . 5 .cod .cod .cod
4240, 41syl5eq 2340 . . . 4 .cod .id .cod .cod
4310, 42syldan 456 . . 3 .Morphism .id .cod .cod
44 cmpidmor2.1 . . . 4
4544, 6, 24, 1cmp2morpcats 26064 . . 3 .id .cod .Morphism .Morphism .id .cod .cod .id .cod .cod .id .cod .id .cod
4636, 38, 39, 43, 45syl121anc 1187 . 2 .Morphism .id .cod .cod .id .cod .id .cod
476, 24, 1, 16morexcmp 26070 . 2 .Morphism .cod
4835, 46, 473eqtr4d 2338 1 .Morphism .id .cod
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   wceq 1632   wcel 1696  cop 3656   cid 4320   cres 4707   ccom 4709  wf 5267  cfv 5271  (class class class)co 5874  c2nd 6137   cmap 6788  cgru 8428  ccmrcase 26013  cdomcase 26022  cgraphcase 26024  ccodcase 26035  cidcase 26042  crocase 26058 This theorem is referenced by:  setiscat  26082 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-map 6790  df-morcatset 26014  df-domcatset 26023  df-graphcatset 26025  df-codcatset 26036  df-idcatset 26043  df-rocatset 26059
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