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Theorem elrnmpti 5531
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 3062 . 2 𝑥𝐴 𝐵 ∈ V
3 rnmpt.1 . . 3 𝐹 = (𝑥𝐴𝐵)
43elrnmptg 5530 . 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 1632  wcel 2139  wral 3050  wrex 3051  Vcvv 3340  cmpt 4881  ran crn 5267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-mpt 4882  df-cnv 5274  df-dm 5276  df-rn 5277
This theorem is referenced by:  fliftel  6722  oarec  7811  unfilem1  8389  pwfilem  8425  elrest  16290  psgneldm2  18124  psgnfitr  18137  iscyggen2  18483  iscyg3  18488  cycsubgcyg  18502  eldprd  18603  leordtval2  21218  iocpnfordt  21221  icomnfordt  21222  lecldbas  21225  tsmsxplem1  22157  minveclem2  23397  lhop2  23977  taylthlem2  24327  fsumvma  25137  dchrptlem2  25189  2sqlem1  25341  dchrisum0fno1  25399  minvecolem2  28040  gsumesum  30430  esumlub  30431  esumcst  30434  esumpcvgval  30449  esumgect  30461  esum2d  30464  sigapildsys  30534  sxbrsigalem2  30657  omssubaddlem  30670  omssubadd  30671  eulerpartgbij  30743  actfunsnf1o  30991  actfunsnrndisj  30992  reprsuc  31002  breprexplema  31017  bnj1366  31207  msubco  31735  msubvrs  31764  fin2so  33709  poimirlem17  33739  poimirlem20  33742  cntotbnd  33908  islsat  34781
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