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Theorem elrnmpt 5404
Description: The range of a function in maps-to notation. (Contributed by Mario Carneiro, 20-Feb-2015.)
Hypothesis
Ref Expression
rnmpt.1 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
elrnmpt (𝐶𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem elrnmpt
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2655 . . 3 (𝑦 = 𝐶 → (𝑦 = 𝐵𝐶 = 𝐵))
21rexbidv 3081 . 2 (𝑦 = 𝐶 → (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
3 rnmpt.1 . . 3 𝐹 = (𝑥𝐴𝐵)
43rnmpt 5403 . 2 ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
52, 4elab2g 3385 1 (𝐶𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1523  wcel 2030  wrex 2942  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-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:  elrnmpt1s  5405  onnseq  7486  oarec  7687  fifo  8379  infpwfien  8923  fin23lem38  9209  fin1a2lem13  9272  ac6num  9339  isercoll2  14443  iserodd  15587  gsumwspan  17430  odf1o2  18034  mplcoe5lem  19515  neitr  21032  ordtbas2  21043  ordtopn1  21046  ordtopn2  21047  pnfnei  21072  mnfnei  21073  pnrmcld  21194  2ndcomap  21309  dis2ndc  21311  ptpjopn  21463  fbasrn  21735  elfm  21798  rnelfmlem  21803  rnelfm  21804  fmfnfmlem3  21807  fmfnfmlem4  21808  fmfnfm  21809  ptcmplem2  21904  tsmsfbas  21978  ustuqtoplem  22090  utopsnneiplem  22098  utopsnnei  22100  utopreg  22103  fmucnd  22143  neipcfilu  22147  imasdsf1olem  22225  xpsdsval  22233  met1stc  22373  metustel  22402  metustsym  22407  metuel2  22417  metustbl  22418  restmetu  22422  xrtgioo  22656  minveclem3b  23245  uniioombllem3  23399  dvivth  23818  gausslemma2dlem1a  25135  elimampt  29566  acunirnmpt  29587  acunirnmpt2  29588  acunirnmpt2f  29589  locfinreflem  30035  ordtconnlem1  30098  esumcst  30253  esumrnmpt2  30258  measdivcstOLD  30415  oms0  30487  omssubadd  30490  cvmsss2  31382  poimirlem16  33555  poimirlem19  33558  poimirlem24  33563  poimirlem27  33566  itg2addnclem2  33592  nelrnmpt  39571  suprnmpt  39669  rnmptpr  39672  elrnmptd  39680  rnmptssrn  39682  wessf1ornlem  39685  disjrnmpt2  39689  disjf1o  39692  disjinfi  39694  choicefi  39706  rnmptlb  39767  rnmptbddlem  39769  rnmptbd2lem  39777  infnsuprnmpt  39779  elmptima  39787  supxrleubrnmpt  39945  suprleubrnmpt  39962  infrnmptle  39963  infxrunb3rnmpt  39968  supminfrnmpt  39985  infxrgelbrnmpt  39996  infrpgernmpt  40008  supminfxrrnmpt  40014  stoweidlem27  40562  stoweidlem31  40566  stoweidlem35  40570  stirlinglem5  40613  stirlinglem13  40621  fourierdlem53  40694  fourierdlem80  40721  fourierdlem93  40734  fourierdlem103  40744  fourierdlem104  40745  subsaliuncllem  40893  subsaliuncl  40894  sge0rnn0  40903  sge00  40911  fsumlesge0  40912  sge0tsms  40915  sge0cl  40916  sge0f1o  40917  sge0fsum  40922  sge0supre  40924  sge0rnbnd  40928  sge0pnffigt  40931  sge0lefi  40933  sge0ltfirp  40935  sge0resplit  40941  sge0split  40944  sge0reuz  40982  sge0reuzb  40983  hoidmvlelem2  41131  smfpimcc  41335
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