Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  dfdm2 Structured version   GIF version

Theorem dfdm2 4795
 Description: Alternate definition of domain df-dm 4298 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.)
Assertion
Ref Expression
dfdm2 dom A = (AA)

Proof of Theorem dfdm2
StepHypRef Expression
1 cnvco 4463 . . . . . 6 (AA) = (AA)
2 cocnvcnv2 4775 . . . . . 6 (AA) = (AA)
31, 2eqtri 2057 . . . . 5 (AA) = (AA)
43unieqi 3581 . . . 4 (AA) = (AA)
54unieqi 3581 . . 3 (AA) = (AA)
6 unidmrn 4793 . . 3 (AA) = (dom (AA) ∪ ran (AA))
75, 6eqtr3i 2059 . 2 (AA) = (dom (AA) ∪ ran (AA))
8 df-rn 4299 . . . . 5 ran A = dom A
98eqcomi 2041 . . . 4 dom A = ran A
10 dmcoeq 4547 . . . 4 (dom A = ran A → dom (AA) = dom A)
119, 10ax-mp 7 . . 3 dom (AA) = dom A
12 rncoeq 4548 . . . . 5 (dom A = ran A → ran (AA) = ran A)
139, 12ax-mp 7 . . . 4 ran (AA) = ran A
14 dfdm4 4470 . . . 4 dom A = ran A
1513, 14eqtr4i 2060 . . 3 ran (AA) = dom A
1611, 15uneq12i 3089 . 2 (dom (AA) ∪ ran (AA)) = (dom A ∪ dom A)
17 unidm 3080 . 2 (dom A ∪ dom A) = dom A
187, 16, 173eqtrri 2062 1 dom A = (AA)
 Colors of variables: wff set class Syntax hints:   = wceq 1242   ∪ cun 2909  ∪ cuni 3571  ◡ccnv 4287  dom cdm 4288  ran crn 4289   ∘ ccom 4292 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator