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

Theorem dfdm2 4952
Description: Alternate definition of domain df-dm 4438 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 4609 . . . . . 6 (𝐴𝐴) = (𝐴𝐴)
2 cocnvcnv2 4929 . . . . . 6 (𝐴𝐴) = (𝐴𝐴)
31, 2eqtri 2108 . . . . 5 (𝐴𝐴) = (𝐴𝐴)
43unieqi 3658 . . . 4 (𝐴𝐴) = (𝐴𝐴)
54unieqi 3658 . . 3 (𝐴𝐴) = (𝐴𝐴)
6 unidmrn 4950 . . 3 (𝐴𝐴) = (dom (𝐴𝐴) ∪ ran (𝐴𝐴))
75, 6eqtr3i 2110 . 2 (𝐴𝐴) = (dom (𝐴𝐴) ∪ ran (𝐴𝐴))
8 df-rn 4439 . . . . 5 ran 𝐴 = dom 𝐴
98eqcomi 2092 . . . 4 dom 𝐴 = ran 𝐴
10 dmcoeq 4693 . . . 4 (dom 𝐴 = ran 𝐴 → dom (𝐴𝐴) = dom 𝐴)
119, 10ax-mp 7 . . 3 dom (𝐴𝐴) = dom 𝐴
12 rncoeq 4694 . . . . 5 (dom 𝐴 = ran 𝐴 → ran (𝐴𝐴) = ran 𝐴)
139, 12ax-mp 7 . . . 4 ran (𝐴𝐴) = ran 𝐴
14 dfdm4 4616 . . . 4 dom 𝐴 = ran 𝐴
1513, 14eqtr4i 2111 . . 3 ran (𝐴𝐴) = dom 𝐴
1611, 15uneq12i 3150 . 2 (dom (𝐴𝐴) ∪ ran (𝐴𝐴)) = (dom 𝐴 ∪ dom 𝐴)
17 unidm 3141 . 2 (dom 𝐴 ∪ dom 𝐴) = dom 𝐴
187, 16, 173eqtrri 2113 1 dom 𝐴 = (𝐴𝐴)
Colors of variables: wff set class
Syntax hints:   = wceq 1289  cun 2995   cuni 3648  ccnv 4427  dom cdm 4428  ran crn 4429  ccom 4432
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator