Theorem rnopab 5921

Description: The range of a class of ordered pairs. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 4-Dec-2016.)

Assertion

Ref	Expression
rnopab	⊢ ran {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑦 ∣ ∃𝑥𝜑}

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem rnopab

Step	Hyp	Ref	Expression
1		nfopab1 5180	. . 3 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
2		nfopab2 5181	. . 3 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
3	1, 2	dfrnf 5917	. 2 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑦 ∣ ∃𝑥 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦}
4		df-br 5111	. . . . 5 ⊢ (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
5		opabidw 5487	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
6	4, 5	bitri 275	. . . 4 ⊢ (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ 𝜑)
7	6	exbii 1848	. . 3 ⊢ (∃𝑥 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ∃𝑥𝜑)
8	7	abbii 2797	. 2 ⊢ {𝑦 ∣ ∃𝑥 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦} = {𝑦 ∣ ∃𝑥𝜑}
9	3, 8	eqtri 2753	1 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑦 ∣ ∃𝑥𝜑}

Syntax hints:   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2708  ⟨cop 4598   class class class wbr 5110  {copab 5172  ran crn 5642

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-cnv 5649  df-dm 5651  df-rn 5652

This theorem is referenced by:  rnopabss  5922  rnopab3  5923  rnmpt  5924  mptpreima  6214  rnoprab  7497  pwfir  9273  marypha2lem4  9396  hartogslem1  9502  rnttrcl  9682  axdc2lem  10408  abrexdomjm  32443  abrexexd  32445  lsmsnorb  33369  satfrnmapom  35364  rnmptsn  37330  abrexdom  37731  rncnvepres  38298  imaopab  42226  tfsconcatrn  43338  modelaxreplem3  44977