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Theorem rncoeq 4548
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 4547 . 2 (dom B = ran A → dom (BA) = dom A)
2 eqcom 2039 . . 3 (dom A = ran B ↔ ran B = dom A)
3 df-rn 4299 . . . 4 ran B = dom B
4 dfdm4 4470 . . . 4 dom A = ran A
53, 4eqeq12i 2050 . . 3 (ran B = dom A ↔ dom B = ran A)
62, 5bitri 173 . 2 (dom A = ran B ↔ dom B = ran A)
7 df-rn 4299 . . . 4 ran (AB) = dom (AB)
8 cnvco 4463 . . . . 5 (AB) = (BA)
98dmeqi 4479 . . . 4 dom (AB) = dom (BA)
107, 9eqtri 2057 . . 3 ran (AB) = dom (BA)
11 df-rn 4299 . . 3 ran A = dom A
1210, 11eqeq12i 2050 . 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 1242  ccnv 4287  dom cdm 4288  ran crn 4289  ccom 4292
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299
This theorem is referenced by:  dfdm2  4795  foco  5059
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