Theorem rnopab 2566

Description: The range of a class of ordered pairs.

Assertion

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

Distinct variable group(s):   x,y

Proof of Theorem rnopab

Step	Hyp	Ref	Expression
1		hbopab1 2112	. . 3 ⊢ (z ∈ {⟨x, y⟩∣φ} → ∀x z ∈ {⟨x, y⟩∣φ})
2		hbopab2 2113	. . 3 ⊢ (z ∈ {⟨x, y⟩∣φ} → ∀y z ∈ {⟨x, y⟩∣φ})
3	1, 2	dfrnf 2561	. 2 ⊢ ran {⟨x, y⟩∣φ} = {y∣∃x⟨x, y⟩ ∈ {⟨x, y⟩∣φ}}
4		opabid 2099	. . . 4 ⊢ (⟨x, y⟩ ∈ {⟨x, y⟩∣φ} ↔ φ)
5	4	biex 733	. . 3 ⊢ (∃x⟨x, y⟩ ∈ {⟨x, y⟩∣φ} ↔ ∃xφ)
6	5	biabi 1181	. 2 ⊢ {y∣∃x⟨x, y⟩ ∈ {⟨x, y⟩∣φ}} = {y∣∃xφ}
7	3, 6	eqtr 1119	1 ⊢ ran {⟨x, y⟩∣φ} = {y∣∃xφ}

Syntax hints:  ∃wex 678  {cab 1090   = wceq 1091   ∈ wcel 1092  ⟨cop 1810  {copab 2055  ran crn 2411

This theorem is referenced by:  fopab2 2891  rnoprab 3033  fo1st 3094  fo2nd 3095  qsex 3231  unfilem1 3438  ac6lem 3575  pjrn 5587

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-cnv 2426  df-dm 2428  df-rn 2429

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