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Theorem cmp2morpgrp 26066
 Description: Graph of the composite of two morphisms. (Contributed by FL, 7-Nov-2013.)
Hypotheses
Ref Expression
cmp2morpcatt.1
cmp2morpcatt.2 .Morphism
cmp2morpcatt.3 .dom
cmp2morpcatt.4 .cod
cmp2morpcatt.5 .graph
Assertion
Ref Expression
cmp2morpgrp .Morphism .Morphism .dom .cod .graph .graph .graph

Proof of Theorem cmp2morpgrp
StepHypRef Expression
1 cmp2morpcatt.1 . . . 4
2 cmp2morpcatt.2 . . . 4 .Morphism
3 cmp2morpcatt.3 . . . 4 .dom
4 cmp2morpcatt.4 . . . 4 .cod
5 eqid 2296 . . . 4
61, 2, 3, 4, 5cmp2morpcatt 26065 . . 3 .Morphism .Morphism .dom .cod .dom .cod
76fveq2d 5545 . 2 .Morphism .Morphism .dom .cod .graph .graph .dom .cod
8 simp1 955 . . 3 .Morphism .Morphism .dom .cod
91, 2, 3, 4rocatval2 26063 . . . 4 .Morphism .Morphism .dom .cod .Morphism
106, 9eqeltrrd 2371 . . 3 .Morphism .Morphism .dom .cod .dom .cod .Morphism
11 cmp2morpcatt.5 . . . 4 .graph
1211, 2isgraphmrph2 26027 . . 3 .dom .cod .Morphism .graph .dom .cod .dom .cod
138, 10, 12syl2anc 642 . 2 .Morphism .Morphism .dom .cod .graph .dom .cod .dom .cod
14 opex 4253 . . . . 5 .dom .cod
15 fvex 5555 . . . . . 6
16 fvex 5555 . . . . . 6
1715, 16coex 5232 . . . . 5
1814, 17op2nd 6145 . . . 4 .dom .cod
1911fveq1i 5542 . . . . . 6 .graph
2019eqcomi 2300 . . . . 5 .graph
2111fveq1i 5542 . . . . . 6 .graph
2221eqcomi 2300 . . . . 5 .graph
2320, 22coeq12i 4863 . . . 4 .graph .graph
2418, 23eqtri 2316 . . 3 .dom .cod .graph .graph
2524a1i 10 . 2 .Morphism .Morphism .dom .cod .dom .cod .graph .graph
267, 13, 253eqtrd 2332 1 .Morphism .Morphism .dom .cod .graph .graph .graph
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   w3a 934   wceq 1632   wcel 1696  cop 3656   ccom 4709  cfv 5271  (class class class)co 5874  c2nd 6137  cgru 8428  ccmrcase 26013  cdomcase 26022  cgraphcase 26024  ccodcase 26035  crocase 26058 This theorem is referenced by:  cmpmorass  26069 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-rocatset 26059
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