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Theorem rncoeq 5974
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 5973 . 2 (dom 𝐵 = ran 𝐴 → dom (𝐵𝐴) = dom 𝐴)
2 eqcom 2733 . . 3 (dom 𝐴 = ran 𝐵 ↔ ran 𝐵 = dom 𝐴)
3 df-rn 5685 . . . 4 ran 𝐵 = dom 𝐵
4 dfdm4 5894 . . . 4 dom 𝐴 = ran 𝐴
53, 4eqeq12i 2744 . . 3 (ran 𝐵 = dom 𝐴 ↔ dom 𝐵 = ran 𝐴)
62, 5bitri 274 . 2 (dom 𝐴 = ran 𝐵 ↔ dom 𝐵 = ran 𝐴)
7 df-rn 5685 . . . 4 ran (𝐴𝐵) = dom (𝐴𝐵)
8 cnvco 5884 . . . . 5 (𝐴𝐵) = (𝐵𝐴)
98dmeqi 5903 . . . 4 dom (𝐴𝐵) = dom (𝐵𝐴)
107, 9eqtri 2754 . . 3 ran (𝐴𝐵) = dom (𝐵𝐴)
11 df-rn 5685 . . 3 ran 𝐴 = dom 𝐴
1210, 11eqeq12i 2744 . 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 1534  ccnv 5673  dom cdm 5674  ran crn 5675  ccom 5678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5146  df-opab 5208  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685
This theorem is referenced by:  dfdm2  6284  algextdeglem4  33593
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