Theorem imaopab 42263

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 ∣ ∃𝑥 ∈ 𝐴 𝜑}

Distinct variable group:   𝑥,𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem imaopab

Step	Hyp	Ref	Expression
1		df-ima 5629	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) = ran ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴)
2		resopab 5983	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
3	2	rneqi 5877	. 2 ⊢ ran ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
4		rnopab 5894	. . 3 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)}
5		df-rex 3057	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
6	5	abbii 2798	. . 3 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)}
7	4, 6	eqtr4i 2757	. 2 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑}
8	1, 3, 7	3eqtri 2758	1 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑}

Syntax hints:   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2111  {cab 2709  ∃wrex 3056  {copab 5153  ran crn 5617   ↾ cres 5618   “ cima 5619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-xp 5622  df-rel 5623  df-cnv 5624  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629

This theorem is referenced by:  prjspeclsp  42644