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

Theorem dfdm2 4880
Description: Alternate definition of domain df-dm 4383 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 4548 . . . . . 6 (𝐴𝐴) = (𝐴𝐴)
2 cocnvcnv2 4860 . . . . . 6 (𝐴𝐴) = (𝐴𝐴)
31, 2eqtri 2076 . . . . 5 (𝐴𝐴) = (𝐴𝐴)
43unieqi 3618 . . . 4 (𝐴𝐴) = (𝐴𝐴)
54unieqi 3618 . . 3 (𝐴𝐴) = (𝐴𝐴)
6 unidmrn 4878 . . 3 (𝐴𝐴) = (dom (𝐴𝐴) ∪ ran (𝐴𝐴))
75, 6eqtr3i 2078 . 2 (𝐴𝐴) = (dom (𝐴𝐴) ∪ ran (𝐴𝐴))
8 df-rn 4384 . . . . 5 ran 𝐴 = dom 𝐴
98eqcomi 2060 . . . 4 dom 𝐴 = ran 𝐴
10 dmcoeq 4632 . . . 4 (dom 𝐴 = ran 𝐴 → dom (𝐴𝐴) = dom 𝐴)
119, 10ax-mp 7 . . 3 dom (𝐴𝐴) = dom 𝐴
12 rncoeq 4633 . . . . 5 (dom 𝐴 = ran 𝐴 → ran (𝐴𝐴) = ran 𝐴)
139, 12ax-mp 7 . . . 4 ran (𝐴𝐴) = ran 𝐴
14 dfdm4 4555 . . . 4 dom 𝐴 = ran 𝐴
1513, 14eqtr4i 2079 . . 3 ran (𝐴𝐴) = dom 𝐴
1611, 15uneq12i 3123 . 2 (dom (𝐴𝐴) ∪ ran (𝐴𝐴)) = (dom 𝐴 ∪ dom 𝐴)
17 unidm 3114 . 2 (dom 𝐴 ∪ dom 𝐴) = dom 𝐴
187, 16, 173eqtrri 2081 1 dom 𝐴 = (𝐴𝐴)
Colors of variables: wff set class
Syntax hints:   = wceq 1259  cun 2943   cuni 3608  ccnv 4372  dom cdm 4373  ran crn 4374  ccom 4377
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator