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Theorem rncoeq 4819
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 4818 . 2 (dom 𝐵 = ran 𝐴 → dom (𝐵𝐴) = dom 𝐴)
2 eqcom 2142 . . 3 (dom 𝐴 = ran 𝐵 ↔ ran 𝐵 = dom 𝐴)
3 df-rn 4557 . . . 4 ran 𝐵 = dom 𝐵
4 dfdm4 4738 . . . 4 dom 𝐴 = ran 𝐴
53, 4eqeq12i 2154 . . 3 (ran 𝐵 = dom 𝐴 ↔ dom 𝐵 = ran 𝐴)
62, 5bitri 183 . 2 (dom 𝐴 = ran 𝐵 ↔ dom 𝐵 = ran 𝐴)
7 df-rn 4557 . . . 4 ran (𝐴𝐵) = dom (𝐴𝐵)
8 cnvco 4731 . . . . 5 (𝐴𝐵) = (𝐵𝐴)
98dmeqi 4747 . . . 4 dom (𝐴𝐵) = dom (𝐵𝐴)
107, 9eqtri 2161 . . 3 ran (𝐴𝐵) = dom (𝐵𝐴)
11 df-rn 4557 . . 3 ran 𝐴 = dom 𝐴
1210, 11eqeq12i 2154 . 2 (ran (𝐴𝐵) = ran 𝐴 ↔ dom (𝐵𝐴) = dom 𝐴)
131, 6, 123imtr4i 200 1 (dom 𝐴 = ran 𝐵 → ran (𝐴𝐵) = ran 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1332  ccnv 4545  dom cdm 4546  ran crn 4547  ccom 4550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-br 3937  df-opab 3997  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557
This theorem is referenced by:  dfdm2  5080  foco  5362
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