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Theorem rncoeq 4900
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 4899 . 2 (dom 𝐵 = ran 𝐴 → dom (𝐵𝐴) = dom 𝐴)
2 eqcom 2179 . . 3 (dom 𝐴 = ran 𝐵 ↔ ran 𝐵 = dom 𝐴)
3 df-rn 4637 . . . 4 ran 𝐵 = dom 𝐵
4 dfdm4 4819 . . . 4 dom 𝐴 = ran 𝐴
53, 4eqeq12i 2191 . . 3 (ran 𝐵 = dom 𝐴 ↔ dom 𝐵 = ran 𝐴)
62, 5bitri 184 . 2 (dom 𝐴 = ran 𝐵 ↔ dom 𝐵 = ran 𝐴)
7 df-rn 4637 . . . 4 ran (𝐴𝐵) = dom (𝐴𝐵)
8 cnvco 4812 . . . . 5 (𝐴𝐵) = (𝐵𝐴)
98dmeqi 4828 . . . 4 dom (𝐴𝐵) = dom (𝐵𝐴)
107, 9eqtri 2198 . . 3 ran (𝐴𝐵) = dom (𝐵𝐴)
11 df-rn 4637 . . 3 ran 𝐴 = dom 𝐴
1210, 11eqeq12i 2191 . 2 (ran (𝐴𝐵) = ran 𝐴 ↔ dom (𝐵𝐴) = dom 𝐴)
131, 6, 123imtr4i 201 1 (dom 𝐴 = ran 𝐵 → ran (𝐴𝐵) = ran 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  ccnv 4625  dom cdm 4626  ran crn 4627  ccom 4630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637
This theorem is referenced by:  dfdm2  5163  foco  5448
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