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Theorem mptima 39936
 Description: Image of a function in map-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 5279 . 2 ((𝑥𝐴𝐵) “ 𝐶) = ran ((𝑥𝐴𝐵) ↾ 𝐶)
2 resmpt3 5608 . . 3 ((𝑥𝐴𝐵) ↾ 𝐶) = (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
32rneqi 5507 . 2 ran ((𝑥𝐴𝐵) ↾ 𝐶) = ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
41, 3eqtri 2782 1 ((𝑥𝐴𝐵) “ 𝐶) = ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1632   ∩ cin 3714   ↦ cmpt 4881  ran crn 5267   ↾ cres 5268   “ cima 5269 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-mpt 4882  df-xp 5272  df-rel 5273  df-cnv 5274  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279 This theorem is referenced by:  mptima2  39956  elmptima  39972
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