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Theorem elrnmptd 5870
Description: The range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
elrnmptd.f 𝐹 = (𝑥𝐴𝐵)
elrnmptd.x (𝜑 → ∃𝑥𝐴 𝐶 = 𝐵)
elrnmptd.c (𝜑𝐶𝑉)
Assertion
Ref Expression
elrnmptd (𝜑𝐶 ∈ ran 𝐹)
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem elrnmptd
StepHypRef Expression
1 elrnmptd.x . 2 (𝜑 → ∃𝑥𝐴 𝐶 = 𝐵)
2 elrnmptd.c . . 3 (𝜑𝐶𝑉)
3 elrnmptd.f . . . 4 𝐹 = (𝑥𝐴𝐵)
43elrnmpt 5865 . . 3 (𝐶𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
52, 4syl 17 . 2 (𝜑 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
61, 5mpbird 256 1 (𝜑𝐶 ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1539  wcel 2106  wrex 3065  cmpt 5157  ran crn 5590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-mpt 5158  df-cnv 5597  df-dm 5599  df-rn 5600
This theorem is referenced by:  pwfilem  8960  nsgmgc  31597  nsgqusf1olem1  31598  zarclsun  31820  rnmptssrn  42719  infnsuprnmpt  42796  supminfrnmpt  42985  supminfxrrnmpt  43011  sge0sup  43929  sge0resplit  43944  sge0xaddlem2  43972  sge0pnfmpt  43983  sge0reuz  43985  sge0reuzb  43986  hoidmvlelem2  44134
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