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Theorem elrnmpti 4892
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 2542 . 2 𝑥𝐴 𝐵 ∈ V
3 rnmpt.1 . . 3 𝐹 = (𝑥𝐴𝐵)
43elrnmptg 4891 . 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 105   = wceq 1363  wcel 2158  wral 2465  wrex 2466  Vcvv 2749  cmpt 4076  ran crn 4639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-mpt 4078  df-cnv 4646  df-dm 4648  df-rn 4649
This theorem is referenced by:  elrest  12712
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