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Theorem rncoeq 5815
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 5814 . 2 (dom 𝐵 = ran 𝐴 → dom (𝐵𝐴) = dom 𝐴)
2 eqcom 2808 . . 3 (dom 𝐴 = ran 𝐵 ↔ ran 𝐵 = dom 𝐴)
3 df-rn 5534 . . . 4 ran 𝐵 = dom 𝐵
4 dfdm4 5732 . . . 4 dom 𝐴 = ran 𝐴
53, 4eqeq12i 2816 . . 3 (ran 𝐵 = dom 𝐴 ↔ dom 𝐵 = ran 𝐴)
62, 5bitri 278 . 2 (dom 𝐴 = ran 𝐵 ↔ dom 𝐵 = ran 𝐴)
7 df-rn 5534 . . . 4 ran (𝐴𝐵) = dom (𝐴𝐵)
8 cnvco 5724 . . . . 5 (𝐴𝐵) = (𝐵𝐴)
98dmeqi 5741 . . . 4 dom (𝐴𝐵) = dom (𝐵𝐴)
107, 9eqtri 2824 . . 3 ran (𝐴𝐵) = dom (𝐵𝐴)
11 df-rn 5534 . . 3 ran 𝐴 = dom 𝐴
1210, 11eqeq12i 2816 . 2 (ran (𝐴𝐵) = ran 𝐴 ↔ dom (𝐵𝐴) = dom 𝐴)
131, 6, 123imtr4i 295 1 (dom 𝐴 = ran 𝐵 → ran (𝐴𝐵) = ran 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  ccnv 5522  dom cdm 5523  ran crn 5524  ccom 5527
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534
This theorem is referenced by:  dfdm2  6104  foco  6581
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