Mathbox for Frédéric Liné < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cmp2morpcats Unicode version

Theorem cmp2morpcats 26064
 Description: Composite of two morphisms. (Contributed by FL, 7-Nov-2013.)
Hypotheses
Ref Expression
cmp2morpcats.1
cmp2morpcats.2 .Morphism
cmp2morpcats.3 .dom
cmp2morpcats.4 .cod
Assertion
Ref Expression
cmp2morpcats .Morphism .Morphism .dom .cod .dom .cod

Proof of Theorem cmp2morpcats
StepHypRef Expression
1 simp1 955 . . 3 .Morphism .Morphism .dom .cod
2 cmp2morpcats.2 . . . . . . 7 .Morphism
32eleq2i 2360 . . . . . 6 .Morphism
43biimpi 186 . . . . 5 .Morphism
54adantr 451 . . . 4 .Morphism .Morphism
653ad2ant2 977 . . 3 .Morphism .Morphism .dom .cod
72eleq2i 2360 . . . . . 6 .Morphism
87biimpi 186 . . . . 5 .Morphism
98adantl 452 . . . 4 .Morphism .Morphism
1093ad2ant2 977 . . 3 .Morphism .Morphism .dom .cod
11 cmp2morpcats.3 . . . . . . 7 .dom
1211fveq1i 5542 . . . . . 6 .dom
13 cmp2morpcats.4 . . . . . . 7 .cod
1413fveq1i 5542 . . . . . 6 .cod
1512, 14eqeq12i 2309 . . . . 5 .dom .cod
1615biimpi 186 . . . 4 .dom .cod
17163ad2ant3 978 . . 3 .Morphism .Morphism .dom .cod
18 cmp2morpcats.1 . . . 4
1918cmp2morp 26061 . . 3
201, 6, 10, 17, 19syl121anc 1187 . 2 .Morphism .Morphism .dom .cod
218ad2antll 709 . . . . . . . . 9 .Morphism .Morphism
22 domcatval 26033 . . . . . . . . 9
2321, 22syldan 456 . . . . . . . 8 .Morphism .Morphism
2423eqcomd 2301 . . . . . . 7 .Morphism .Morphism
25 fveq1 5540 . . . . . . . 8 .dom .dom
2625eqeq2d 2307 . . . . . . 7 .dom .dom
2724, 26syl5ibr 212 . . . . . 6 .dom .Morphism .Morphism .dom
2811, 27ax-mp 8 . . . . 5 .Morphism .Morphism .dom
29283adant3 975 . . . 4 .Morphism .Morphism .dom .cod .dom
305anim2i 552 . . . . . . . . 9 .Morphism .Morphism
31303adant3 975 . . . . . . . 8 .Morphism .Morphism .dom .cod
32 codcatval 26039 . . . . . . . 8
3331, 32syl 15 . . . . . . 7 .Morphism .Morphism .dom .cod
34 fveq1 5540 . . . . . . . 8 .cod .cod
3534eqeq1d 2304 . . . . . . 7 .cod .cod
3633, 35syl5ibr 212 . . . . . 6 .cod .Morphism .Morphism .dom .cod .cod
3713, 36ax-mp 8 . . . . 5 .Morphism .Morphism .dom .cod .cod
3837eqcomd 2301 . . . 4 .Morphism .Morphism .dom .cod .cod
3929, 38opeq12d 3820 . . 3 .Morphism .Morphism .dom .cod .dom .cod
4039opeq1d 3818 . 2 .Morphism .Morphism .dom .cod .dom .cod
4120, 40eqtrd 2328 1 .Morphism .Morphism .dom .cod .dom .cod
 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  c1st 6136  c2nd 6137  cgru 8428  ccmrcase 26013  cdomcase 26022  ccodcase 26035  crocase 26058 This theorem is referenced by:  cmp2morpcatt  26065  cmp2morpdom  26067  cmp2morpcod  26068  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-mpt2 5879  df-1st 6138  df-2nd 6139  df-domcatset 26023  df-codcatset 26036  df-rocatset 26059
 Copyright terms: Public domain W3C validator