Theorem rnoprab 6743

Description: The range of an operation class abstraction. (Contributed by NM, 30-Aug-2004.) (Revised by David Abernethy, 19-Apr-2013.)

Assertion

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

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem rnoprab

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfoprab2 6701	. . 3 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2	1	rneqi 5352	. 2 ⊢ ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = ran {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
3		rnopab 5370	. 2 ⊢ ran {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑧 ∣ ∃𝑤∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
4		exrot3 2045	. . . 4 ⊢ (∃𝑤∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦∃𝑤(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
5		opex 4932	. . . . . . 7 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
6	5	isseti 3209	. . . . . 6 ⊢ ∃𝑤 𝑤 = ⟨𝑥, 𝑦⟩
7		19.41v 1914	. . . . . 6 ⊢ (∃𝑤(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (∃𝑤 𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
8	6, 7	mpbiran 953	. . . . 5 ⊢ (∃𝑤(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜑)
9	8	2exbii 1775	. . . 4 ⊢ (∃𝑥∃𝑦∃𝑤(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦𝜑)
10	4, 9	bitri 264	. . 3 ⊢ (∃𝑤∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦𝜑)
11	10	abbii 2739	. 2 ⊢ {𝑧 ∣ ∃𝑤∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑧 ∣ ∃𝑥∃𝑦𝜑}
12	2, 3, 11	3eqtri 2648	1 ⊢ ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦𝜑}

Syntax hints:   ∧ wa 384   = wceq 1483  ∃wex 1704  {cab 2608  ⟨cop 4183  {copab 4712  ran crn 5115  {coprab 6651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-cnv 5122  df-dm 5124  df-rn 5125  df-oprab 6654

This theorem is referenced by:  rnoprab2  6744  elrnmpt2res  6774  ellines  32259