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Theorem elmptima 45102
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 6100 . . . 4 ((𝑥𝐴𝐵) “ 𝐷) = ran (𝑥 ∈ (𝐴𝐷) ↦ 𝐵)
21a1i 11 . . 3 (𝐶𝑉 → ((𝑥𝐴𝐵) “ 𝐷) = ran (𝑥 ∈ (𝐴𝐷) ↦ 𝐵))
32eleq2d 2824 . 2 (𝐶𝑉 → (𝐶 ∈ ((𝑥𝐴𝐵) “ 𝐷) ↔ 𝐶 ∈ ran (𝑥 ∈ (𝐴𝐷) ↦ 𝐵)))
4 eqid 2734 . . 3 (𝑥 ∈ (𝐴𝐷) ↦ 𝐵) = (𝑥 ∈ (𝐴𝐷) ↦ 𝐵)
54elrnmpt 5980 . 2 (𝐶𝑉 → (𝐶 ∈ ran (𝑥 ∈ (𝐴𝐷) ↦ 𝐵) ↔ ∃𝑥 ∈ (𝐴𝐷)𝐶 = 𝐵))
63, 5bitrd 279 1 (𝐶𝑉 → (𝐶 ∈ ((𝑥𝐴𝐵) “ 𝐷) ↔ ∃𝑥 ∈ (𝐴𝐷)𝐶 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wcel 2103  wrex 3072  cin 3969  cmpt 5252  ran crn 5700  cima 5702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5170  df-opab 5232  df-mpt 5253  df-xp 5705  df-rel 5706  df-cnv 5707  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712
This theorem is referenced by:  liminfvalxr  45639
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