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

Theorem dfdm2 5175
Description: Alternate definition of domain df-dm 4648 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 4824 . . . . . 6 (𝐴𝐴) = (𝐴𝐴)
2 cocnvcnv2 5152 . . . . . 6 (𝐴𝐴) = (𝐴𝐴)
31, 2eqtri 2208 . . . . 5 (𝐴𝐴) = (𝐴𝐴)
43unieqi 3831 . . . 4 (𝐴𝐴) = (𝐴𝐴)
54unieqi 3831 . . 3 (𝐴𝐴) = (𝐴𝐴)
6 unidmrn 5173 . . 3 (𝐴𝐴) = (dom (𝐴𝐴) ∪ ran (𝐴𝐴))
75, 6eqtr3i 2210 . 2 (𝐴𝐴) = (dom (𝐴𝐴) ∪ ran (𝐴𝐴))
8 df-rn 4649 . . . . 5 ran 𝐴 = dom 𝐴
98eqcomi 2191 . . . 4 dom 𝐴 = ran 𝐴
10 dmcoeq 4911 . . . 4 (dom 𝐴 = ran 𝐴 → dom (𝐴𝐴) = dom 𝐴)
119, 10ax-mp 5 . . 3 dom (𝐴𝐴) = dom 𝐴
12 rncoeq 4912 . . . . 5 (dom 𝐴 = ran 𝐴 → ran (𝐴𝐴) = ran 𝐴)
139, 12ax-mp 5 . . . 4 ran (𝐴𝐴) = ran 𝐴
14 dfdm4 4831 . . . 4 dom 𝐴 = ran 𝐴
1513, 14eqtr4i 2211 . . 3 ran (𝐴𝐴) = dom 𝐴
1611, 15uneq12i 3299 . 2 (dom (𝐴𝐴) ∪ ran (𝐴𝐴)) = (dom 𝐴 ∪ dom 𝐴)
17 unidm 3290 . 2 (dom 𝐴 ∪ dom 𝐴) = dom 𝐴
187, 16, 173eqtrri 2213 1 dom 𝐴 = (𝐴𝐴)
Colors of variables: wff set class
Syntax hints:   = wceq 1363  cun 3139   cuni 3821  ccnv 4637  dom cdm 4638  ran crn 4639  ccom 4642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator