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Theorem rncoeq 4694
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 4693 . 2 (dom 𝐵 = ran 𝐴 → dom (𝐵𝐴) = dom 𝐴)
2 eqcom 2090 . . 3 (dom 𝐴 = ran 𝐵 ↔ ran 𝐵 = dom 𝐴)
3 df-rn 4439 . . . 4 ran 𝐵 = dom 𝐵
4 dfdm4 4616 . . . 4 dom 𝐴 = ran 𝐴
53, 4eqeq12i 2101 . . 3 (ran 𝐵 = dom 𝐴 ↔ dom 𝐵 = ran 𝐴)
62, 5bitri 182 . 2 (dom 𝐴 = ran 𝐵 ↔ dom 𝐵 = ran 𝐴)
7 df-rn 4439 . . . 4 ran (𝐴𝐵) = dom (𝐴𝐵)
8 cnvco 4609 . . . . 5 (𝐴𝐵) = (𝐵𝐴)
98dmeqi 4625 . . . 4 dom (𝐴𝐵) = dom (𝐵𝐴)
107, 9eqtri 2108 . . 3 ran (𝐴𝐵) = dom (𝐵𝐴)
11 df-rn 4439 . . 3 ran 𝐴 = dom 𝐴
1210, 11eqeq12i 2101 . 2 (ran (𝐴𝐵) = ran 𝐴 ↔ dom (𝐵𝐴) = dom 𝐴)
131, 6, 123imtr4i 199 1 (dom 𝐴 = ran 𝐵 → ran (𝐴𝐵) = ran 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  ccnv 4427  dom cdm 4428  ran crn 4429  ccom 4432
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439
This theorem is referenced by:  dfdm2  4952  foco  5227
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