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Theorem arwval 16016
 Description: The set of arrows is the union of all the disjointified hom-sets. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
arwval.a Nat
arwval.h Homa
Assertion
Ref Expression
arwval

Proof of Theorem arwval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 arwval.a . 2 Nat
2 fveq2 5879 . . . . . . 7 Homa Homa
3 arwval.h . . . . . . 7 Homa
42, 3syl6eqr 2523 . . . . . 6 Homa
54rneqd 5068 . . . . 5 Homa
65unieqd 4200 . . . 4 Homa
7 df-arw 16000 . . . 4 Nat Homa
8 fvex 5889 . . . . . . 7 Homa
93, 8eqeltri 2545 . . . . . 6
109rnex 6746 . . . . 5
1110uniex 6606 . . . 4
126, 7, 11fvmpt 5963 . . 3 Nat
137dmmptss 5338 . . . . . . 7 Nat
1413sseli 3414 . . . . . 6 Nat
1514con3i 142 . . . . 5 Nat
16 ndmfv 5903 . . . . 5 Nat Nat
1715, 16syl 17 . . . 4 Nat
18 df-homa 15999 . . . . . . . . . . . . 13 Homa
1918dmmptss 5338 . . . . . . . . . . . 12 Homa
2019sseli 3414 . . . . . . . . . . 11 Homa
2120con3i 142 . . . . . . . . . 10 Homa
22 ndmfv 5903 . . . . . . . . . 10 Homa Homa
2321, 22syl 17 . . . . . . . . 9 Homa
243, 23syl5eq 2517 . . . . . . . 8
2524rneqd 5068 . . . . . . 7
26 rn0 5092 . . . . . . 7
2725, 26syl6eq 2521 . . . . . 6
2827unieqd 4200 . . . . 5
29 uni0 4217 . . . . 5
3028, 29syl6eq 2521 . . . 4
3117, 30eqtr4d 2508 . . 3 Nat
3212, 31pm2.61i 169 . 2 Nat
331, 32eqtri 2493 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wceq 1452   wcel 1904  cvv 3031  c0 3722  csn 3959  cuni 4190   cmpt 4454   cxp 4837   cdm 4839   crn 4840  cfv 5589  cbs 15199   chom 15279  ccat 15648  Natcarw 15995  Homachoma 15996 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fv 5597  df-homa 15999  df-arw 16000 This theorem is referenced by:  arwhoma  16018  homarw  16019
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