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Theorem mptima 6061
Description: Image of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Assertion
Ref Expression
mptima ((𝑥𝐴𝐵) “ 𝐶) = ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem mptima
StepHypRef Expression
1 df-ima 5679 . 2 ((𝑥𝐴𝐵) “ 𝐶) = ran ((𝑥𝐴𝐵) ↾ 𝐶)
2 resmpt3 6028 . . 3 ((𝑥𝐴𝐵) ↾ 𝐶) = (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
32rneqi 5926 . 2 ran ((𝑥𝐴𝐵) ↾ 𝐶) = ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
41, 3eqtri 2752 1 ((𝑥𝐴𝐵) “ 𝐶) = ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  cin 3939  cmpt 5221  ran crn 5667  cres 5668  cima 5669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-mpt 5222  df-xp 5672  df-rel 5673  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679
This theorem is referenced by:  mptimass  6062  fsplitfpar  8098  elmptima  44447
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