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Theorem mptima 39259
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 5125 . 2 ((𝑥𝐴𝐵) “ 𝐶) = ran ((𝑥𝐴𝐵) ↾ 𝐶)
2 resmpt3 5448 . . 3 ((𝑥𝐴𝐵) ↾ 𝐶) = (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
32rneqi 5350 . 2 ran ((𝑥𝐴𝐵) ↾ 𝐶) = ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
41, 3eqtri 2643 1 ((𝑥𝐴𝐵) “ 𝐶) = ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1482  cin 3571  cmpt 4727  ran crn 5113  cres 5114  cima 5115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-br 4652  df-opab 4711  df-mpt 4728  df-xp 5118  df-rel 5119  df-cnv 5120  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125
This theorem is referenced by:  mptima2  39279  elmptima  39295
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