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Theorem rncoeq 4532
 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 4531 . 2 (dom B = ran A → dom (BA) = dom A)
2 eqcom 2024 . . 3 (dom A = ran B ↔ ran B = dom A)
3 df-rn 4283 . . . 4 ran B = dom B
4 dfdm4 4454 . . . 4 dom A = ran A
53, 4eqeq12i 2035 . . 3 (ran B = dom A ↔ dom B = ran A)
62, 5bitri 173 . 2 (dom A = ran B ↔ dom B = ran A)
7 df-rn 4283 . . . 4 ran (AB) = dom (AB)
8 cnvco 4447 . . . . 5 (AB) = (BA)
98dmeqi 4463 . . . 4 dom (AB) = dom (BA)
107, 9eqtri 2042 . . 3 ran (AB) = dom (BA)
11 df-rn 4283 . . 3 ran A = dom A
1210, 11eqeq12i 2035 . 2 (ran (AB) = ran A ↔ dom (BA) = dom A)
131, 6, 123imtr4i 190 1 (dom A = ran B → ran (AB) = ran A)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1228  ◡ccnv 4271  dom cdm 4272  ran crn 4273   ∘ ccom 4276 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283 This theorem is referenced by:  dfdm2  4779  foco  5041
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