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

Theorem rnmptc 7227
Description: Range of a constant function in maps-to notation. (Contributed by Glauco Siliprandi, 11-Dec-2019.) Remove extra hypothesis. (Revised by SN, 17-Apr-2024.)
Hypotheses
Ref Expression
rnmptc.f 𝐹 = (𝑥𝐴𝐵)
rnmptc.a (𝜑𝐴 ≠ ∅)
Assertion
Ref Expression
rnmptc (𝜑 → ran 𝐹 = {𝐵})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝐹(𝑥)

Proof of Theorem rnmptc
StepHypRef Expression
1 rnmptc.a . 2 (𝜑𝐴 ≠ ∅)
2 rnmptc.f . . . . 5 𝐹 = (𝑥𝐴𝐵)
3 fconstmpt 5747 . . . . 5 (𝐴 × {𝐵}) = (𝑥𝐴𝐵)
42, 3eqtr4i 2768 . . . 4 𝐹 = (𝐴 × {𝐵})
54rneqi 5948 . . 3 ran 𝐹 = ran (𝐴 × {𝐵})
6 rnxp 6190 . . 3 (𝐴 ≠ ∅ → ran (𝐴 × {𝐵}) = {𝐵})
75, 6eqtrid 2789 . 2 (𝐴 ≠ ∅ → ran 𝐹 = {𝐵})
81, 7syl 17 1 (𝜑 → ran 𝐹 = {𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wne 2940  c0 4333  {csn 4626  cmpt 5225   × cxp 5683  ran crn 5686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-mpt 5226  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696
This theorem is referenced by:  mptiffisupp  32702  qsalrel  42281  limsup0  45709  limsuppnfdlem  45716  limsup10ex  45788  liminf10ex  45789  fourierdlem60  46181  fourierdlem61  46182  sge0z  46390
  Copyright terms: Public domain W3C validator