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Theorem morexcmp 26070
 Description: A morphism expressed thanks to its components. (Contributed by FL, 8-Nov-2013.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
Hypotheses
Ref Expression
morexcmp.1 .Morphism
morexcmp.2 .dom
morexcmp.3 .cod
morexcmp.4 .graph
Assertion
Ref Expression
morexcmp .Morphism .dom .cod .graph

Proof of Theorem morexcmp
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 morexcmp.1 . . . . . 6 .Morphism
2 morcatset1 26018 . . . . . 6
31, 2syl5eq 2340 . . . . 5 .Morphism
4 oprabss 5949 . . . . . 6
54a1i 10 . . . . 5
63, 5eqsstrd 3225 . . . 4 .Morphism
76sselda 3193 . . 3 .Morphism
8 1st2nd2 6175 . . 3
97, 8syl 15 . 2 .Morphism
10 xp1st 6165 . . . . 5
11 1st2nd2 6175 . . . . 5
127, 10, 113syl 18 . . . 4 .Morphism
13 morexcmp.2 . . . . . . 7 .dom
141, 13domcatval2 26034 . . . . . 6 .Morphism .dom
15 fo1st 6155 . . . . . . . 8
16 fof 5467 . . . . . . . 8
1715, 16ax-mp 8 . . . . . . 7
18 elex 2809 . . . . . . . 8 .Morphism
1918adantl 452 . . . . . . 7 .Morphism
20 fvco3 5612 . . . . . . 7
2117, 19, 20sylancr 644 . . . . . 6 .Morphism
2214, 21eqtr2d 2329 . . . . 5 .Morphism .dom
23 morexcmp.3 . . . . . . 7 .cod
241, 23codcatval2 26040 . . . . . 6 .Morphism .cod
25 fvco3 5612 . . . . . . 7
2617, 19, 25sylancr 644 . . . . . 6 .Morphism
2724, 26eqtr2d 2329 . . . . 5 .Morphism .cod
2822, 27opeq12d 3820 . . . 4 .Morphism .dom .cod
2912, 28eqtrd 2328 . . 3 .Morphism .dom .cod
30 morexcmp.4 . . . . . 6 .graph
3130eqcomi 2300 . . . . 5 .graph
3231fveq1i 5542 . . . 4 .graph
3332a1i 10 . . 3 .Morphism .graph
3429, 33opeq12d 3820 . 2 .Morphism .dom .cod .graph
359, 34eqtrd 2328 1 .Morphism .dom .cod .graph
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   w3a 934   wceq 1632   wcel 1696  cvv 2801   wss 3165  cop 3656   cxp 4703   ccom 4709  wf 5267  wfo 5269  cfv 5271  (class class class)co 5874  coprab 5875  c1st 6136  c2nd 6137   cmap 6788  cgru 8428  ccmrcase 26013  cdomcase 26022  ccodcase 26035 This theorem is referenced by:  morexcmp2  26071  cmpidmor2  26072 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-1st 6138  df-2nd 6139  df-morcatset 26014  df-domcatset 26023  df-codcatset 26036
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