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

Theorem rncoeq 5930
Description: Range of a composition. (Contributed by NM, 19-Mar-1998.)
Assertion
Ref Expression
rncoeq (dom 𝐴 = ran 𝐵 → ran (𝐴𝐵) = ran 𝐴)

Proof of Theorem rncoeq
StepHypRef Expression
1 dmcoeq 5929 . 2 (dom 𝐵 = ran 𝐴 → dom (𝐵𝐴) = dom 𝐴)
2 eqcom 2743 . . 3 (dom 𝐴 = ran 𝐵 ↔ ran 𝐵 = dom 𝐴)
3 df-rn 5644 . . . 4 ran 𝐵 = dom 𝐵
4 dfdm4 5851 . . . 4 dom 𝐴 = ran 𝐴
53, 4eqeq12i 2754 . . 3 (ran 𝐵 = dom 𝐴 ↔ dom 𝐵 = ran 𝐴)
62, 5bitri 274 . 2 (dom 𝐴 = ran 𝐵 ↔ dom 𝐵 = ran 𝐴)
7 df-rn 5644 . . . 4 ran (𝐴𝐵) = dom (𝐴𝐵)
8 cnvco 5841 . . . . 5 (𝐴𝐵) = (𝐵𝐴)
98dmeqi 5860 . . . 4 dom (𝐴𝐵) = dom (𝐵𝐴)
107, 9eqtri 2764 . . 3 ran (𝐴𝐵) = dom (𝐵𝐴)
11 df-rn 5644 . . 3 ran 𝐴 = dom 𝐴
1210, 11eqeq12i 2754 . 2 (ran (𝐴𝐵) = ran 𝐴 ↔ dom (𝐵𝐴) = dom 𝐴)
131, 6, 123imtr4i 291 1 (dom 𝐴 = ran 𝐵 → ran (𝐴𝐵) = ran 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  ccnv 5632  dom cdm 5633  ran crn 5634  ccom 5637
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-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-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644
This theorem is referenced by:  dfdm2  6233
  Copyright terms: Public domain W3C validator