Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  arwval Structured version   Unicode version

Theorem arwval 14200
 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 5730 . . . . . . 7 Homa Homa
3 arwval.h . . . . . . 7 Homa
42, 3syl6eqr 2488 . . . . . 6 Homa
54rneqd 5099 . . . . 5 Homa
65unieqd 4028 . . . 4 Homa
7 df-arw 14184 . . . 4 Nat Homa
8 fvex 5744 . . . . . . 7 Homa
93, 8eqeltri 2508 . . . . . 6
109rnex 5135 . . . . 5
1110uniex 4707 . . . 4
126, 7, 11fvmpt 5808 . . 3 Nat
137dmmptss 5368 . . . . . . 7 Nat
1413sseli 3346 . . . . . 6 Nat
1514con3i 130 . . . . 5 Nat
16 ndmfv 5757 . . . . 5 Nat Nat
1715, 16syl 16 . . . 4 Nat
18 df-homa 14183 . . . . . . . . . . . . 13 Homa
1918dmmptss 5368 . . . . . . . . . . . 12 Homa
2019sseli 3346 . . . . . . . . . . 11 Homa
2120con3i 130 . . . . . . . . . 10 Homa
22 ndmfv 5757 . . . . . . . . . 10 Homa Homa
2321, 22syl 16 . . . . . . . . 9 Homa
243, 23syl5eq 2482 . . . . . . . 8
2524rneqd 5099 . . . . . . 7
26 rn0 5129 . . . . . . 7
2725, 26syl6eq 2486 . . . . . 6
2827unieqd 4028 . . . . 5
29 uni0 4044 . . . . 5
3028, 29syl6eq 2486 . . . 4
3117, 30eqtr4d 2473 . . 3 Nat
3212, 31pm2.61i 159 . 2 Nat
331, 32eqtri 2458 1
 Colors of variables: wff set class Syntax hints:   wn 3   wceq 1653   wcel 1726  cvv 2958  c0 3630  csn 3816  cuni 4017   cmpt 4268   cxp 4878   cdm 4880   crn 4881  cfv 5456  cbs 13471   chom 13542  ccat 13891  Natcarw 14179  Homachoma 14180 This theorem is referenced by:  arwhoma  14202  homarw  14203 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fv 5464  df-homa 14183  df-arw 14184
 Copyright terms: Public domain W3C validator