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Theorem elrnmpti 5408
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 2953 . 2 𝑥𝐴 𝐵 ∈ V
3 rnmpt.1 . . 3 𝐹 = (𝑥𝐴𝐵)
43elrnmptg 5407 . 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 196   = wceq 1523  wcel 2030  wral 2941  wrex 2942  Vcvv 3231  cmpt 4762  ran crn 5144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-mpt 4763  df-cnv 5151  df-dm 5153  df-rn 5154
This theorem is referenced by:  fliftel  6599  oarec  7687  unfilem1  8265  pwfilem  8301  elrest  16135  psgneldm2  17970  psgnfitr  17983  iscyggen2  18329  iscyg3  18334  cycsubgcyg  18348  eldprd  18449  leordtval2  21064  iocpnfordt  21067  icomnfordt  21068  lecldbas  21071  tsmsxplem1  22003  minveclem2  23243  lhop2  23823  taylthlem2  24173  fsumvma  24983  dchrptlem2  25035  2sqlem1  25187  dchrisum0fno1  25245  minvecolem2  27859  gsumesum  30249  esumlub  30250  esumcst  30253  esumpcvgval  30268  esumgect  30280  esum2d  30283  sigapildsys  30353  sxbrsigalem2  30476  omssubaddlem  30489  omssubadd  30490  eulerpartgbij  30562  actfunsnf1o  30810  actfunsnrndisj  30811  reprsuc  30821  breprexplema  30836  bnj1366  31026  msubco  31554  msubvrs  31583  fin2so  33526  poimirlem17  33556  poimirlem20  33559  cntotbnd  33725  islsat  34596
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