Theorem imaopab 42725

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 5638	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) = ran ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴)
2		resopab 5993	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
3	2	rneqi 5886	. 2 ⊢ ran ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
4		rnopab 5903	. . 3 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)}
5		df-rex 3065	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
6	5	abbii 2807	. . 3 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)}
7	4, 6	eqtr4i 2766	. 2 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑}
8	1, 3, 7	3eqtri 2767	1 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑}

Syntax hints:   ∧ wa 396   = wceq 1547  ∃wex 1786   ∈ wcel 2119  {cab 2718  ∃wrex 3064  {copab 5141  ran crn 5626   ↾ cres 5627   “ cima 5628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-xp 5631  df-rel 5632  df-cnv 5633  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638

This theorem is referenced by:  prjspeclsp  43069