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Theorem rncoeq 4632
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 4631 . 2 (dom 𝐵 = ran 𝐴 → dom (𝐵𝐴) = dom 𝐴)
2 eqcom 2058 . . 3 (dom 𝐴 = ran 𝐵 ↔ ran 𝐵 = dom 𝐴)
3 df-rn 4383 . . . 4 ran 𝐵 = dom 𝐵
4 dfdm4 4554 . . . 4 dom 𝐴 = ran 𝐴
53, 4eqeq12i 2069 . . 3 (ran 𝐵 = dom 𝐴 ↔ dom 𝐵 = ran 𝐴)
62, 5bitri 177 . 2 (dom 𝐴 = ran 𝐵 ↔ dom 𝐵 = ran 𝐴)
7 df-rn 4383 . . . 4 ran (𝐴𝐵) = dom (𝐴𝐵)
8 cnvco 4547 . . . . 5 (𝐴𝐵) = (𝐵𝐴)
98dmeqi 4563 . . . 4 dom (𝐴𝐵) = dom (𝐵𝐴)
107, 9eqtri 2076 . . 3 ran (𝐴𝐵) = dom (𝐵𝐴)
11 df-rn 4383 . . 3 ran 𝐴 = dom 𝐴
1210, 11eqeq12i 2069 . 2 (ran (𝐴𝐵) = ran 𝐴 ↔ dom (𝐵𝐴) = dom 𝐴)
131, 6, 123imtr4i 194 1 (dom 𝐴 = ran 𝐵 → ran (𝐴𝐵) = ran 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  ccnv 4371  dom cdm 4372  ran crn 4373  ccom 4376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383
This theorem is referenced by:  dfdm2  4879  foco  5143
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