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Theorem imaopab 39920
Description: The image of a class of ordered pairs. (Contributed by Steven Nguyen, 6-Jun-2023.)
Assertion
Ref Expression
imaopab ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝜑}
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem imaopab
StepHypRef Expression
1 df-ima 5564 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) = ran ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴)
2 resopab 5902 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
32rneqi 5806 . 2 ran ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
4 rnopab 5823 . . 3 ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = {𝑦 ∣ ∃𝑥(𝑥𝐴𝜑)}
5 df-rex 3067 . . . 4 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
65abbii 2808 . . 3 {𝑦 ∣ ∃𝑥𝐴 𝜑} = {𝑦 ∣ ∃𝑥(𝑥𝐴𝜑)}
74, 6eqtr4i 2768 . 2 ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = {𝑦 ∣ ∃𝑥𝐴 𝜑}
81, 3, 73eqtri 2769 1 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝜑}
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1543  wex 1787  wcel 2110  {cab 2714  wrex 3062  {copab 5115  ran crn 5552  cres 5553  cima 5554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-xp 5557  df-rel 5558  df-cnv 5559  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564
This theorem is referenced by:  prjspeclsp  40159
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