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Theorem elrnmpti 5283
Description: Membership in the range of a function. (Contributed by NM, 30-Aug-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypotheses
Ref Expression
rnmpt.1 𝐹 = (𝑥𝐴𝐵)
elrnmpti.2 𝐵 ∈ V
Assertion
Ref Expression
elrnmpti (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵)
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem elrnmpti
StepHypRef Expression
1 elrnmpti.2 . . 3 𝐵 ∈ V
21rgenw 2907 . 2 𝑥𝐴 𝐵 ∈ V
3 rnmpt.1 . . 3 𝐹 = (𝑥𝐴𝐵)
43elrnmptg 5282 . 2 (∀𝑥𝐴 𝐵 ∈ V → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
52, 4ax-mp 5 1 (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 194   = wceq 1474  wcel 1976  wral 2895  wrex 2896  Vcvv 3172  cmpt 4637  ran crn 5028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-mpt 4639  df-cnv 5035  df-dm 5037  df-rn 5038
This theorem is referenced by:  fliftel  6436  oarec  7506  unfilem1  8086  pwfilem  8120  elrest  15859  psgneldm2  17695  psgnfitr  17708  iscyggen2  18054  iscyg3  18059  cycsubgcyg  18073  eldprd  18174  leordtval2  20773  iocpnfordt  20776  icomnfordt  20777  lecldbas  20780  tsmsxplem1  21713  minveclem2  22949  lhop2  23526  taylthlem2  23876  fsumvma  24682  dchrptlem2  24734  2sqlem1  24886  dchrisum0fno1  24944  minvecolem2  26908  gsumesum  29241  esumlub  29242  esumcst  29245  esumpcvgval  29260  esumgect  29272  esum2d  29275  sigapildsys  29345  sxbrsigalem2  29468  omssubaddlem  29481  omssubadd  29482  eulerpartgbij  29554  bnj1366  29947  msubco  30475  msubvrs  30504  fin2so  32349  poimirlem17  32379  poimirlem20  32382  cntotbnd  32548  islsat  33079
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