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Theorem imaopab 41356
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 5688 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) = ran ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴)
2 resopab 6033 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
32rneqi 5935 . 2 ran ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
4 rnopab 5952 . . 3 ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = {𝑦 ∣ ∃𝑥(𝑥𝐴𝜑)}
5 df-rex 3069 . . . 4 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
65abbii 2800 . . 3 {𝑦 ∣ ∃𝑥𝐴 𝜑} = {𝑦 ∣ ∃𝑥(𝑥𝐴𝜑)}
74, 6eqtr4i 2761 . 2 ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = {𝑦 ∣ ∃𝑥𝐴 𝜑}
81, 3, 73eqtri 2762 1 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝜑}
Colors of variables: wff setvar class
Syntax hints:  wa 394   = wceq 1539  wex 1779  wcel 2104  {cab 2707  wrex 3068  {copab 5209  ran crn 5676  cres 5677  cima 5678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-cnv 5683  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688
This theorem is referenced by:  prjspeclsp  41656
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