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Theorem mptima 5913
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 5540 . 2 ((𝑥𝐴𝐵) “ 𝐶) = ran ((𝑥𝐴𝐵) ↾ 𝐶)
2 resmpt3 5878 . . 3 ((𝑥𝐴𝐵) ↾ 𝐶) = (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
32rneqi 5779 . 2 ran ((𝑥𝐴𝐵) ↾ 𝐶) = ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
41, 3eqtri 2843 1 ((𝑥𝐴𝐵) “ 𝐶) = ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  cin 3908  cmpt 5118  ran crn 5528  cres 5529  cima 5530
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5175  ax-nul 5182  ax-pr 5302
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-rab 3134  df-v 3472  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5039  df-opab 5101  df-mpt 5119  df-xp 5533  df-rel 5534  df-cnv 5535  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540
This theorem is referenced by:  fsplitfpar  7788  mptima2  41667  elmptima  41680
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