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Theorem elmptima 41821
 Description: The image of a function in maps-to notation. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Assertion
Ref Expression
elmptima (𝐶𝑉 → (𝐶 ∈ ((𝑥𝐴𝐵) “ 𝐷) ↔ ∃𝑥 ∈ (𝐴𝐷)𝐶 = 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷
Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem elmptima
StepHypRef Expression
1 mptima 5928 . . . 4 ((𝑥𝐴𝐵) “ 𝐷) = ran (𝑥 ∈ (𝐴𝐷) ↦ 𝐵)
21a1i 11 . . 3 (𝐶𝑉 → ((𝑥𝐴𝐵) “ 𝐷) = ran (𝑥 ∈ (𝐴𝐷) ↦ 𝐵))
32eleq2d 2901 . 2 (𝐶𝑉 → (𝐶 ∈ ((𝑥𝐴𝐵) “ 𝐷) ↔ 𝐶 ∈ ran (𝑥 ∈ (𝐴𝐷) ↦ 𝐵)))
4 eqid 2824 . . 3 (𝑥 ∈ (𝐴𝐷) ↦ 𝐵) = (𝑥 ∈ (𝐴𝐷) ↦ 𝐵)
54elrnmpt 5815 . 2 (𝐶𝑉 → (𝐶 ∈ ran (𝑥 ∈ (𝐴𝐷) ↦ 𝐵) ↔ ∃𝑥 ∈ (𝐴𝐷)𝐶 = 𝐵))
63, 5bitrd 282 1 (𝐶𝑉 → (𝐶 ∈ ((𝑥𝐴𝐵) “ 𝐷) ↔ ∃𝑥 ∈ (𝐴𝐷)𝐶 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2115  ∃wrex 3134   ∩ cin 3918   ↦ cmpt 5132  ran crn 5543   “ cima 5545 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-mpt 5133  df-xp 5548  df-rel 5549  df-cnv 5550  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555 This theorem is referenced by:  liminfvalxr  42351
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