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Theorem cmp2morpdom 25964
 Description: Domain of the composite of two morphisms. (Contributed by FL, 7-Nov-2013.) (Revised by Mario Carneiro, 27-Dec-2014.)
Hypotheses
Ref Expression
cmp2morpcatt.1
cmp2morpcatt.2 .Morphism
cmp2morpcatt.3 .dom
cmp2morpcatt.4 .cod
Assertion
Ref Expression
cmp2morpdom .Morphism .Morphism .dom .cod .dom .dom

Proof of Theorem cmp2morpdom
StepHypRef Expression
1 cmp2morpcatt.3 . . . 4 .dom
21a1i 10 . . 3 .Morphism .Morphism .dom .cod .dom
32fveq1d 5527 . 2 .Morphism .Morphism .dom .cod .dom
4 simp1 955 . . 3 .Morphism .Morphism .dom .cod
5 cmp2morpcatt.2 . . . . . . 7 .Morphism
65eleq2i 2347 . . . . . 6 .Morphism
76biimpi 186 . . . . 5 .Morphism
85eleq2i 2347 . . . . . 6 .Morphism
98biimpi 186 . . . . 5 .Morphism
107, 9anim12i 549 . . . 4 .Morphism .Morphism
11 cmp2morpcatt.1 . . . . 5
12 eqid 2283 . . . . 5
13 cmp2morpcatt.4 . . . . 5 .cod
1411, 12, 1, 13rocatval2 25960 . . . 4 .dom .cod
1510, 14syl3an2 1216 . . 3 .Morphism .Morphism .dom .cod
16 domcatval 25930 . . 3
174, 15, 16syl2anc 642 . 2 .Morphism .Morphism .dom .cod
1811, 5, 1, 13cmp2morpcats 25961 . . . 4 .Morphism .Morphism .dom .cod .dom .cod
1918fveq2d 5529 . . 3 .Morphism .Morphism .dom .cod .dom .cod
20 fo1st 6139 . . . . . 6
21 fof 5451 . . . . . 6
2220, 21ax-mp 8 . . . . 5
23 opex 4237 . . . . 5 .dom .cod
24 fvco3 5596 . . . . 5 .dom .cod .dom .cod .dom .cod
2522, 23, 24mp2an 653 . . . 4 .dom .cod .dom .cod
26 opex 4237 . . . . . 6 .dom .cod
27 fvex 5539 . . . . . . 7
28 fvex 5539 . . . . . . 7
2927, 28coex 5216 . . . . . 6
3026, 29op1st 6128 . . . . 5 .dom .cod .dom .cod
3130fveq2i 5528 . . . 4 .dom .cod .dom .cod
32 fvex 5539 . . . . 5 .dom
33 fvex 5539 . . . . 5 .cod
3432, 33op1st 6128 . . . 4 .dom .cod .dom
3525, 31, 343eqtri 2307 . . 3 .dom .cod .dom
3619, 35syl6eq 2331 . 2 .Morphism .Morphism .dom .cod .dom
373, 17, 363eqtrd 2319 1 .Morphism .Morphism .dom .cod .dom .dom
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   w3a 934   wceq 1623   wcel 1684  cvv 2788  cop 3643   ccom 4693  wf 5251  wfo 5253  cfv 5255  (class class class)co 5858  c1st 6120  c2nd 6121  cgru 8412  ccmrcase 25910  cdomcase 25919  ccodcase 25932  crocase 25955 This theorem is referenced by:  cmpmorass  25966  setiscat  25979 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-morcatset 25911  df-domcatset 25920  df-codcatset 25933  df-rocatset 25956
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