MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elrnmptd Structured version   Visualization version   GIF version

Theorem elrnmptd 5960
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 5955 . . 3 (𝐶𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
52, 4syl 17 . 2 (𝜑 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
61, 5mpbird 257 1 (𝜑𝐶 ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1540  wcel 2105  wrex 3069  cmpt 5231  ran crn 5677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-mpt 5232  df-cnv 5684  df-dm 5686  df-rn 5687
This theorem is referenced by:  pwfilem  9180  nsgmgc  32798  nsgqusf1olem1  32799  evls1maprnss  33051  algextdeglem4  33066  zarclsun  33149  rnmptssrn  44180  infnsuprnmpt  44253  supminfrnmpt  44454  supminfxrrnmpt  44480  sge0sup  45406  sge0resplit  45421  sge0xaddlem2  45449  sge0pnfmpt  45460  sge0reuz  45462  sge0reuzb  45463  hoidmvlelem2  45611
  Copyright terms: Public domain W3C validator