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

Theorem mptimass 6091
Description: Image of a function in maps-to notation for a subset. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypothesis
Ref Expression
mptimass.1 (𝜑𝐶𝐴)
Assertion
Ref Expression
mptimass (𝜑 → ((𝑥𝐴𝐵) “ 𝐶) = ran (𝑥𝐶𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem mptimass
StepHypRef Expression
1 mptima 6090 . 2 ((𝑥𝐴𝐵) “ 𝐶) = ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
2 mptimass.1 . . . . 5 (𝜑𝐶𝐴)
3 sseqin2 4223 . . . . 5 (𝐶𝐴 ↔ (𝐴𝐶) = 𝐶)
42, 3sylib 218 . . . 4 (𝜑 → (𝐴𝐶) = 𝐶)
54mpteq1d 5237 . . 3 (𝜑 → (𝑥 ∈ (𝐴𝐶) ↦ 𝐵) = (𝑥𝐶𝐵))
65rneqd 5949 . 2 (𝜑 → ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵) = ran (𝑥𝐶𝐵))
71, 6eqtrid 2789 1 (𝜑 → ((𝑥𝐴𝐵) “ 𝐶) = ran (𝑥𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  cin 3950  wss 3951  cmpt 5225  ran crn 5686  cima 5688
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-10 2141  ax-12 2177  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-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  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  df-res 5697  df-ima 5698
This theorem is referenced by:  pzriprnglem10  21501  limsupresico  45715  limsupvaluz  45723  liminfresico  45786
  Copyright terms: Public domain W3C validator