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Theorem rncoeq 4975
 Description: Range of a composition. (Contributed by set.mm contributors, 19-Mar-1998.)
Assertion
Ref Expression
rncoeq (dom A = ran B → ran (A B) = ran A)

Proof of Theorem rncoeq
StepHypRef Expression
1 dmcoeq 4974 . 2 (dom B = ran A → dom (B A) = dom A)
2 df-dm 4787 . . . 4 dom A = ran A
3 dfrn4 4904 . . . 4 ran B = dom B
42, 3eqeq12i 2366 . . 3 (dom A = ran B ↔ ran A = dom B)
5 eqcom 2355 . . 3 (ran A = dom B ↔ dom B = ran A)
64, 5bitri 240 . 2 (dom A = ran B ↔ dom B = ran A)
7 dfrn4 4904 . . . 4 ran (A B) = dom (A B)
8 cnvco 4894 . . . . 5 (A B) = (B A)
98dmeqi 4908 . . . 4 dom (A B) = dom (B A)
107, 9eqtri 2373 . . 3 ran (A B) = dom (B A)
11 dfrn4 4904 . . 3 ran A = dom A
1210, 11eqeq12i 2366 . 2 (ran (A B) = ran A ↔ dom (B A) = dom A)
131, 6, 123imtr4i 257 1 (dom A = ran B → ran (A B) = ran A)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ∘ ccom 4721  ◡ccnv 4771  dom cdm 4772  ran crn 4773 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-cnv 4785  df-rn 4786  df-dm 4787 This theorem is referenced by:  foco  5279
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