Theorem rnoprab 5862

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 5826	. . 3 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2	1	rneqi 4775	. 2 ⊢ ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = ran {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
3		rnopab 4794	. 2 ⊢ ran {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑧 ∣ ∃𝑤∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
4		exrot3 1669	. . . 4 ⊢ (∃𝑤∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦∃𝑤(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
5		vex 2692	. . . . . . . 8 ⊢ 𝑥 ∈ V
6		vex 2692	. . . . . . . 8 ⊢ 𝑦 ∈ V
7	5, 6	opex 4159	. . . . . . 7 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
8	7	isseti 2697	. . . . . 6 ⊢ ∃𝑤 𝑤 = ⟨𝑥, 𝑦⟩
9		19.41v 1875	. . . . . 6 ⊢ (∃𝑤(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (∃𝑤 𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
10	8, 9	mpbiran 925	. . . . 5 ⊢ (∃𝑤(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜑)
11	10	2exbii 1586	. . . 4 ⊢ (∃𝑥∃𝑦∃𝑤(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦𝜑)
12	4, 11	bitri 183	. . 3 ⊢ (∃𝑤∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦𝜑)
13	12	abbii 2256	. 2 ⊢ {𝑧 ∣ ∃𝑤∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑧 ∣ ∃𝑥∃𝑦𝜑}
14	2, 3, 13	3eqtri 2165	1 ⊢ ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦𝜑}

Syntax hints:   ∧ wa 103   = wceq 1332  ∃wex 1469  {cab 2126  ⟨cop 3535  {copab 3996  ran crn 4548  {coprab 5783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-cnv 4555  df-dm 4557  df-rn 4558  df-oprab 5786

This theorem is referenced by:  rnoprab2  5863