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Theorem elrnmpti 4799
 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 2490 . 2 𝑥𝐴 𝐵 ∈ V
3 rnmpt.1 . . 3 𝐹 = (𝑥𝐴𝐵)
43elrnmptg 4798 . 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 set class Syntax hints:   ↔ wb 104   = wceq 1332   ∈ wcel 1481  ∀wral 2417  ∃wrex 2418  Vcvv 2689   ↦ cmpt 3996  ran crn 4547 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-br 3937  df-opab 3997  df-mpt 3998  df-cnv 4554  df-dm 4556  df-rn 4557 This theorem is referenced by:  elrest  12164
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