Theorem rnopab 5935

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 5175	. . 3 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
2		nfopab2 5176	. . 3 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
3	1, 2	dfrnf 5931	. 2 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑦 ∣ ∃𝑥 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦}
4		df-br 5106	. . . . 5 ⊢ (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
5		opabidw 5499	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
6	4, 5	bitri 278	. . . 4 ⊢ (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ 𝜑)
7	6	exbii 1871	. . 3 ⊢ (∃𝑥 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ∃𝑥𝜑)
8	7	abbii 2832	. 2 ⊢ {𝑦 ∣ ∃𝑥 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦} = {𝑦 ∣ ∃𝑥𝜑}
9	3, 8	eqtri 2788	1 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑦 ∣ ∃𝑥𝜑}

Syntax hints:   = wceq 1563  ∃wex 1802   ∈ wcel 2145  {cab 2743  ⟨cop 4591   class class class wbr 5105  {copab 5167  ran crn 5653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-cnv 5660  df-dm 5662  df-rn 5663

This theorem is referenced by:  rnopabss  5936  rnopab3  5937  rnmpt  5938  mptpreima  6229  rnoprab  7505  pwfir  9264  marypha2lem4  9386  hartogslem1  9492  rnttrcl  9679  axdc2lem  10420  abrexdomjm  32763  abrexexd  32765  lsmsnorb  33620  satfrnmapom  35733  rnmptsn  37841  abrexdom  38241  rncnvepres  38820  dfsuccl2  38981  imaopab  42862  tfsconcatrn  43931  modelaxreplem3  45554