MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  imadisj Structured version   Visualization version   GIF version

Theorem imadisj 6032
Description: A class whose image under another is empty is disjoint with the other's domain. (Contributed by FL, 24-Jan-2007.)
Assertion
Ref Expression
imadisj ((𝐴𝐵) = ∅ ↔ (dom 𝐴𝐵) = ∅)

Proof of Theorem imadisj
StepHypRef Expression
1 df-ima 5646 . . 3 (𝐴𝐵) = ran (𝐴𝐵)
21eqeq1i 2741 . 2 ((𝐴𝐵) = ∅ ↔ ran (𝐴𝐵) = ∅)
3 dm0rn0 5880 . 2 (dom (𝐴𝐵) = ∅ ↔ ran (𝐴𝐵) = ∅)
4 dmres 5959 . . . 4 dom (𝐴𝐵) = (𝐵 ∩ dom 𝐴)
5 incom 4161 . . . 4 (𝐵 ∩ dom 𝐴) = (dom 𝐴𝐵)
64, 5eqtri 2764 . . 3 dom (𝐴𝐵) = (dom 𝐴𝐵)
76eqeq1i 2741 . 2 (dom (𝐴𝐵) = ∅ ↔ (dom 𝐴𝐵) = ∅)
82, 3, 73bitr2i 298 1 ((𝐴𝐵) = ∅ ↔ (dom 𝐴𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1541  cin 3909  c0 4282  dom cdm 5633  ran crn 5634  cres 5635  cima 5636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-br 5106  df-opab 5168  df-xp 5639  df-cnv 5641  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646
This theorem is referenced by:  ndmima  6055  fnimadisj  6633  fnimaeq0  6634  fimacnvdisj  6720  frxp2  8075  frxp3  8082  acndom2  9989  isf34lem5  10313  isf34lem7  10314  isf34lem6  10315  limsupgre  15362  isercolllem3  15550  pf1rcl  21713  cnconn  22771  1stcfb  22794  xkohaus  23002  qtopeu  23065  fbasrn  23233  mbflimsup  25028  preiman0  31568  eulerpartlemt  32911  erdszelem5  33729  fnwe2lem2  41355  imadisjld  42413  imadisjlnd  42414
  Copyright terms: Public domain W3C validator