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Theorem rncoeq 5354
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 5353 . 2 (dom 𝐵 = ran 𝐴 → dom (𝐵𝐴) = dom 𝐴)
2 eqcom 2628 . . 3 (dom 𝐴 = ran 𝐵 ↔ ran 𝐵 = dom 𝐴)
3 df-rn 5090 . . . 4 ran 𝐵 = dom 𝐵
4 dfdm4 5281 . . . 4 dom 𝐴 = ran 𝐴
53, 4eqeq12i 2635 . . 3 (ran 𝐵 = dom 𝐴 ↔ dom 𝐵 = ran 𝐴)
62, 5bitri 264 . 2 (dom 𝐴 = ran 𝐵 ↔ dom 𝐵 = ran 𝐴)
7 df-rn 5090 . . . 4 ran (𝐴𝐵) = dom (𝐴𝐵)
8 cnvco 5273 . . . . 5 (𝐴𝐵) = (𝐵𝐴)
98dmeqi 5290 . . . 4 dom (𝐴𝐵) = dom (𝐵𝐴)
107, 9eqtri 2643 . . 3 ran (𝐴𝐵) = dom (𝐵𝐴)
11 df-rn 5090 . . 3 ran 𝐴 = dom 𝐴
1210, 11eqeq12i 2635 . 2 (ran (𝐴𝐵) = ran 𝐴 ↔ dom (𝐵𝐴) = dom 𝐴)
131, 6, 123imtr4i 281 1 (dom 𝐴 = ran 𝐵 → ran (𝐴𝐵) = ran 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  ccnv 5078  dom cdm 5079  ran crn 5080  ccom 5083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090
This theorem is referenced by:  dfdm2  5631  foco  6087
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