Theorem rnopab 5894

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 5161	. . 3 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
2		nfopab2 5162	. . 3 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
3	1, 2	dfrnf 5890	. 2 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑦 ∣ ∃𝑥 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦}
4		df-br 5092	. . . . 5 ⊢ (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
5		opabidw 5464	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
6	4, 5	bitri 275	. . . 4 ⊢ (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ 𝜑)
7	6	exbii 1849	. . 3 ⊢ (∃𝑥 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ∃𝑥𝜑)
8	7	abbii 2798	. 2 ⊢ {𝑦 ∣ ∃𝑥 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦} = {𝑦 ∣ ∃𝑥𝜑}
9	3, 8	eqtri 2754	1 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑦 ∣ ∃𝑥𝜑}

Syntax hints:   = wceq 1541  ∃wex 1780   ∈ wcel 2111  {cab 2709  ⟨cop 4582   class class class wbr 5091  {copab 5153  ran crn 5617

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-rab 3396  df-v 3438  df-dif 3905  df-un 3907  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-cnv 5624  df-dm 5626  df-rn 5627

This theorem is referenced by:  rnopabss  5895  rnopab3  5896  rnmpt  5897  mptpreima  6185  rnoprab  7451  pwfir  9201  marypha2lem4  9322  hartogslem1  9428  rnttrcl  9612  axdc2lem  10336  abrexdomjm  32482  abrexexd  32484  lsmsnorb  33351  satfrnmapom  35402  rnmptsn  37368  abrexdom  37769  rncnvepres  38336  imaopab  42263  tfsconcatrn  43374  modelaxreplem3  45012