Theorem rnoprab 3033

Description: The range of an operation class abstraction.

Assertion

Ref	Expression
rnoprab	⊢ ran {⟨⟨x, y⟩, z⟩∣φ} = {z∣∃x∃yφ}

Distinct variable group(s):   x,y,z

Proof of Theorem rnoprab

Step	Hyp	Ref	Expression
1		dfoprab2 3021	. . 3 ⊢ {⟨⟨x, y⟩, z⟩∣φ} = {⟨w, z⟩∣∃x∃y(w = ⟨x, y⟩ ∧ φ)}
2	1	rneqi 2556	. 2 ⊢ ran {⟨⟨x, y⟩, z⟩∣φ} = ran {⟨w, z⟩∣∃x∃y(w = ⟨x, y⟩ ∧ φ)}
3		rnopab 2566	. 2 ⊢ ran {⟨w, z⟩∣∃x∃y(w = ⟨x, y⟩ ∧ φ)} = {z∣∃w∃x∃y(w = ⟨x, y⟩ ∧ φ)}
4		exrot3 777	. . . 4 ⊢ (∃w∃x∃y(w = ⟨x, y⟩ ∧ φ) ↔ ∃x∃y∃w(w = ⟨x, y⟩ ∧ φ))
5		19.41v 963	. . . . . 6 ⊢ (∃w(w = ⟨x, y⟩ ∧ φ) ↔ (∃w w = ⟨x, y⟩ ∧ φ))
6		opex 1893	. . . . . . 7 ⊢ ⟨x, y⟩ ∈ V
7	6	isseti 1352	. . . . . 6 ⊢ ∃w w = ⟨x, y⟩
8	5, 7	mpbiran 547	. . . . 5 ⊢ (∃w(w = ⟨x, y⟩ ∧ φ) ↔ φ)
9	8	bi2ex 734	. . . 4 ⊢ (∃x∃y∃w(w = ⟨x, y⟩ ∧ φ) ↔ ∃x∃yφ)
10	4, 9	bitr 151	. . 3 ⊢ (∃w∃x∃y(w = ⟨x, y⟩ ∧ φ) ↔ ∃x∃yφ)
11	10	biabi 1181	. 2 ⊢ {z∣∃w∃x∃y(w = ⟨x, y⟩ ∧ φ)} = {z∣∃x∃yφ}
12	2, 3, 11	3eqtr 1123	1 ⊢ ran {⟨⟨x, y⟩, z⟩∣φ} = {z∣∃x∃yφ}

Syntax hints:   ∧ wa 196  ∃wex 678  {cab 1090   = wceq 1091  ⟨cop 1810  {copab 2055  ran crn 2411  {copab2 3002

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  df-oprab 3004

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