Theorem rnoprab 7465

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 7418	. . 3 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2	1	rneqi 5886	. 2 ⊢ ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = ran {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
3		rnopab 5903	. 2 ⊢ ran {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑧 ∣ ∃𝑤∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
4		exrot3 2171	. . . 4 ⊢ (∃𝑤∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦∃𝑤(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
5		opex 5411	. . . . . . 7 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
6	5	isseti 3448	. . . . . 6 ⊢ ∃𝑤 𝑤 = ⟨𝑥, 𝑦⟩
7		19.41v 1951	. . . . . 6 ⊢ (∃𝑤(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (∃𝑤 𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
8	6, 7	mpbiran 710	. . . . 5 ⊢ (∃𝑤(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜑)
9	8	2exbii 1851	. . . 4 ⊢ (∃𝑥∃𝑦∃𝑤(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦𝜑)
10	4, 9	bitri 275	. . 3 ⊢ (∃𝑤∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦𝜑)
11	10	abbii 2804	. 2 ⊢ {𝑧 ∣ ∃𝑤∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑧 ∣ ∃𝑥∃𝑦𝜑}
12	2, 3, 11	3eqtri 2764	1 ⊢ ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦𝜑}

Syntax hints:   ∧ wa 395   = wceq 1542  ∃wex 1781  {cab 2715  ⟨cop 4574  {copab 5148  ran crn 5625  {coprab 7361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-cnv 5632  df-dm 5634  df-rn 5635  df-oprab 7364

This theorem is referenced by:  rnoprab2  7466  elrnmpores  7498  ellines  36350  dmxrn  38722