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Theorem rnex 5107
 Description: The range of a set is a set. Corollary 6.8(3) of [TakeutiZaring] p. 26. Similar to Lemma 3D of [Enderton] p. 41. (Contributed by set.mm contributors, 7-Jul-2008.)
Hypothesis
Ref Expression
dmex.1 A V
Assertion
Ref Expression
rnex ran A V

Proof of Theorem rnex
StepHypRef Expression
1 dmex.1 . 2 A V
2 rnexg 5104 . 2 (A V → ran A V)
31, 2ax-mp 5 1 ran A V
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 1710  Vcvv 2859  ran crn 4773 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  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-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  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-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-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-br 4640  df-ima 4727  df-rn 4786 This theorem is referenced by:  elxp4  5108  ffoss  5314  foundex  5914  mapexi  6003  bren  6030  enex  6031  enmap1lem5  6073  ovmuc  6130  mucex  6133  ovcelem1  6171  ceex  6174  sbthlem1  6203  sbthlem3  6205  tcfnex  6244  nmembers1lem1  6268  nncdiv3lem2  6276  nnc3n3p1  6278  nchoicelem11  6299  nchoicelem16  6304  nchoicelem18  6306  frecxp  6314
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