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Theorem prismorcset3 26041
 Description: The predicate "is a morphism of the category Set". (Contributed by FL, 6-Nov-2013.)
Hypotheses
Ref Expression
prismorcset3.1 .dom
prismorcset3.2 .cod
prismorcset3.3 .graph
prismorcset3.4 .Morphism
Assertion
Ref Expression
prismorcset3 .Morphism .graph .cod .dom

Proof of Theorem prismorcset3
StepHypRef Expression
1 prismorcset3.4 . 2 .Morphism
2 eleq2 2357 . . . 4 .Morphism .Morphism
32anbi2d 684 . . 3 .Morphism .Morphism
4 prismorcset3.1 . . . 4 .dom
5 prismorcset3.2 . . . . . 6 .cod
6 prismorcset3.3 . . . . . . . 8 .graph
7 eqid 2296 . . . . . . . . . . . 12
8 eqid 2296 . . . . . . . . . . . 12
9 eqid 2296 . . . . . . . . . . . 12
107, 8, 9prismorcset2 26021 . . . . . . . . . . 11
1110simp3d 969 . . . . . . . . . 10
12 isgraphmrph 26026 . . . . . . . . . . 11
13 codcatval 26039 . . . . . . . . . . . 12
14 domcatval 26033 . . . . . . . . . . . 12
1513, 14oveq12d 5892 . . . . . . . . . . 11
1612, 15eleq12d 2364 . . . . . . . . . 10
1711, 16mpbird 223 . . . . . . . . 9
18 fveq1 5540 . . . . . . . . . 10 .graph .graph
1918eleq1d 2362 . . . . . . . . 9 .graph .graph
2017, 19syl5ibr 212 . . . . . . . 8 .graph .graph
216, 20ax-mp 8 . . . . . . 7 .graph
22 fveq1 5540 . . . . . . . . 9 .cod .cod
2322oveq1d 5889 . . . . . . . 8 .cod .cod
2423eleq2d 2363 . . . . . . 7 .cod .graph .cod .graph
2521, 24syl5ibr 212 . . . . . 6 .cod .graph .cod
265, 25ax-mp 8 . . . . 5 .graph .cod
27 fveq1 5540 . . . . . . 7 .dom .dom
2827oveq2d 5890 . . . . . 6 .dom .cod .dom .cod
2928eleq2d 2363 . . . . 5 .dom .graph .cod .dom .graph .cod
3026, 29syl5ibr 212 . . . 4 .dom .graph .cod .dom
314, 30ax-mp 8 . . 3 .graph .cod .dom
323, 31syl6bi 219 . 2 .Morphism .Morphism .graph .cod .dom
331, 32ax-mp 8 1 .Morphism .graph .cod .dom
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   wceq 1632   wcel 1696   ccom 4709  cfv 5271  (class class class)co 5874  c1st 6136  c2nd 6137   cmap 6788  cgru 8428  ccmrcase 26013  cdomcase 26022  cgraphcase 26024  ccodcase 26035 This theorem is referenced by:  cmpidmor2  26072  cmpidmor3  26073 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-graphcatset 26025  df-codcatset 26036
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