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Theorem mptimass 6093
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 6092 . 2 ((𝑥𝐴𝐵) “ 𝐶) = ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
2 mptimass.1 . . . . 5 (𝜑𝐶𝐴)
3 sseqin2 4231 . . . . 5 (𝐶𝐴 ↔ (𝐴𝐶) = 𝐶)
42, 3sylib 218 . . . 4 (𝜑 → (𝐴𝐶) = 𝐶)
54mpteq1d 5243 . . 3 (𝜑 → (𝑥 ∈ (𝐴𝐶) ↦ 𝐵) = (𝑥𝐶𝐵))
65rneqd 5952 . 2 (𝜑 → ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵) = ran (𝑥𝐶𝐵))
71, 6eqtrid 2787 1 (𝜑 → ((𝑥𝐴𝐵) “ 𝐶) = ran (𝑥𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  cin 3962  wss 3963  cmpt 5231  ran crn 5690  cima 5692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702
This theorem is referenced by:  pzriprnglem10  21519  limsupresico  45656  limsupvaluz  45664  liminfresico  45727
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