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Theorem imaopab 40864
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 5682 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) = ran ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴)
2 resopab 6024 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
32rneqi 5928 . 2 ran ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
4 rnopab 5945 . . 3 ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = {𝑦 ∣ ∃𝑥(𝑥𝐴𝜑)}
5 df-rex 3070 . . . 4 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
65abbii 2801 . . 3 {𝑦 ∣ ∃𝑥𝐴 𝜑} = {𝑦 ∣ ∃𝑥(𝑥𝐴𝜑)}
74, 6eqtr4i 2762 . 2 ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = {𝑦 ∣ ∃𝑥𝐴 𝜑}
81, 3, 73eqtri 2763 1 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝜑}
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1541  wex 1781  wcel 2106  {cab 2708  wrex 3069  {copab 5203  ran crn 5670  cres 5671  cima 5672
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682
This theorem is referenced by:  prjspeclsp  41134
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