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Theorem imaopab 39433
 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 5533 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) = ran ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴)
2 resopab 5870 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
32rneqi 5772 . 2 ran ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
4 rnopab 5791 . . 3 ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = {𝑦 ∣ ∃𝑥(𝑥𝐴𝜑)}
5 df-rex 3112 . . . 4 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
65abbii 2863 . . 3 {𝑦 ∣ ∃𝑥𝐴 𝜑} = {𝑦 ∣ ∃𝑥(𝑥𝐴𝜑)}
74, 6eqtr4i 2824 . 2 ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = {𝑦 ∣ ∃𝑥𝐴 𝜑}
81, 3, 73eqtri 2825 1 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝜑}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2776  ∃wrex 3107  {copab 5093  ran crn 5521   ↾ cres 5522   “ cima 5523 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5032  df-opab 5094  df-xp 5526  df-rel 5527  df-cnv 5528  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533 This theorem is referenced by:  prjspeclsp  39649
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